
Class TT 5-^ i 
Book ■\N t(n 



THE 



PROPORTIONS AND MOVEMENT 



OF 



SLIDE VALVES. 



BY 



WILLIAM DYSON WANSBROUGH, 

If 

Author of 
" The A B C of the Differential Calculus," 
The Portable Engine ; its Construction and Management," 
etc., etc. 



Price 4s. 6d. Net. 



1903. 

The Technical Publishing Co., Ltd., Manchester, 

AND ALL Booksellers. 






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LC Control Number 




tmp96 027065 



PEEFACE. 



Some explanation of the re-appearance of these articles 
in book form, so long after their original publication, may be 
offered. 

From time to time many requests for re-pubhcation have 
reached the author, both directly and through the publishers, 
from users of the articles, who felt the inconvenience of 
tracing the particular case required through a series of trade 
numbers. 

The Author's thanks are due to the Proprietors of The 
Mechanical World, in which the series originally appeared, 
for their courtesy, which has enabled him to collect and 
reprint the articles in what he hopes will prove a con- 
venient and acceptable form for reference. 
Lincoln, March, 1903. 



CONTENTS : 

PAGES 

Introduction 1 — 7 

Chapter I. Cases 1—13 8—22 

Chapter II. Analysis. Cases 14 — 21 ... 22—38 

Chapter III. The Double Slide Valve.— 

Series A. Cases 22— 24 38-64 

Chapter IV. Expansion Plate, to cut off with 
its inner edges, and having no lap, either 
positive or negative. — Series B. Cases 
30—35 ... . 64-87 

Chapter V. Expansion Plate, to cut off with 
its outer edges, and having negative or 
minuslap.— Series C, Cases 36— 44 ... 87—121 

Chapter VI. The Meyer Cut-off gear. — Series 

DandE. Cases 45— 50 122—160 



INDEX TO CASES. 



Chapter I. 

PAGE 

Case I. — Lap, lead, and cut-off being given : To find 

the throw and angle of the eccentric 8 

Case II. — Lap, lead, and travel being given : To find 

the angle of the eccentric and the point of cut-off 8 

Case III. — The throw of the eccentric, the lap, and the 
point of cut-off being given : To find the lead 
and the angle of the eccentric 9 

Case IV. — The throw of the eccentric, the lead, and 
the point of cut-off being given : To find the 
lap and the angle of the eccentric ... ... 10 

Case V. — The throw and angle of the eccentric, and the 
lead being given : To find the lap and the 
point of cut-off 11 

Case VI. — The throw of the eccentric, the lead, and a 
position of the crank where the steam port is 
to be open a certain fixed amount being given : 
To find the lap, the angle of the eccentric, and 
the point of cut-off... ... ... ... ... 11 

Case VII. — The lap and the lead of the valve, and a 
position of the crank where the steam port is 
to be open a certain fixed amount being given : 
To find the throw and the angle of the eccentric 
and the point of cut-off 12 

Case VIII. — The lead, the point of cut-off, and a position 
of the crank where the steam port is to be 
open a certain fixed amount being given : To 
find the lap, the throw of the eccentric, and its 
angle with the crank 13 

Case IX. — The lead, the maximum opening of the port, 
and the point of cut-off being given : To find 
the lap, the throw of the eccentric, and its 
angle with the crank 15 

Case X. — Variable cut-off by a simple slide valve with 
a constant lead, lap, lead, and the points of 
cut-off being given : To find the several throws 
and angular positions of the adjustable 
pccentric 16 



INDEX TO CASES. 



PAGE 



Case XI.— In a central admission valve : Lap, lead, and 
cut-off being given : To find the throw of the 
eccentric and its angle with the crank 18 

Case XII.— ExliailSt ; Throw and angle of eccentric 
being given: To find the points of release and 
closure of the exhaust 19 

Case XIII.— Exhaust Lap ! Throw and angle of 
eccentric being given : To find the exhaust lap 
(if any) required to release the steam at a 
given position of the crank. ... ... ... 20 



Chapter II. — Analysis. 

Case XIV. — Lap of valve, throw and angle of eccentric 
being given : To find the points of admission 
and cat-off; and the position of the crank or of 
the piston, when the port is open a given 
amount 23 

Case XV. — Lap, throw, and angle of eccentric being 
given : To find the amount the steam port is 
open for a given position of the piston or of 
the crank ... ... ... ... ... ... 24 

Case XVI. — Lap, throw, and angle of eccentric being 
given; To find all the points in the cycle of 
operations for one revolution of the crank, the 
slide valve having neither exhaust Jap nor 
clearance 24 

Case XVII. — Lap, throw, and angle of eccentric being 
given to find all the points in the cycle of 
operations for one revolution of the crank, a 
given amount of exhaust lap being added to the 
inner edges of the slide valve ... ,.. ... 28 

Case XVIII. — Variable cut-off by a simple slide valve, 
with a constant lead, lap, lead, and the points 
of cut-off being given, and the several throws 
and angular positions of the adjustable 
eccentric being laid down as in Case X. : To 
construct a Zeuner diagram, there being 
neither exhaust lap nor clearance 30 



INDEX TO CASES. 



PAGE 



Case XIX.— Valves with Central Admission (as 

piston valves, see Case XI.). Lap, throw, and 
angle of eccentric being given : To find all the 
points in the cycle of operations for one revolu- 
tion of the crank 32 

Case XX.— An old-fashioned Slide Valve; To 

construct the diagram from given dimensions... 35 

Case XXI.— A suggested improvement in the delineation 
of the exhaust port opening in the Zeuner 
diagram 36 

Chapter III. — Double Slide Valve. 

Case XXII.— To construct the main and expansion 
eccentrics for a given cylinder face ; the lap of 
the main valve, the lead, the point of cut off, 
the width of the slot, and a position of the 
crank where a certain opening is desired, being 
given 41 

Case XXIII. — Conditions as in Case XXII. To construct 
the Zeuner diagram and to delineate the 
curve of port opening (steamway) ... ... 43 

Case XXIV.' — Conditions as in Case XXII, but with 
double or multiple-slott<ed expansion plate. 
To construct the valves and eccentrics for a 
given cylinder face ; the lap of main valve, the 
lead, the point of cut-off, the width of slots and 
a position of the crank where a total slot open- 
ing is desired, being given ... 45 

Case XXV. — Conditions as in Case XXIV. To construct 
the Zeuner diagram and from it the valve and 
expansion plate ; and to delineate the curves of 
port opening (steamway) for diagrams 24, 24a, 
and 24b 50 

Case XXVI. — Variable expansion by altering the angle 
of the eccentric, its throw being fixed. Particu- 
lars of main valve and its eccentric being 
given, itogether with the maximum travel 
permissible for the expansion plate : To find 
the respective angles of the expansion eccentric 
for cutting off at given positions of the crank 
or piston 57 



INDEX TO CASES. IX. 

PAGE 

Case XXVII. — Variable expansion by altering the angle 
of the eccentric, its throw being fixed. Con- 
ditions as in Case XXVI : To construct the 
Zeuner diagram, and to delineate the curves of 
port opening (steamway) for the several points 
of cut-off 58 

Case XXVIII.— Variable expansion by altering the 
throw of the eccentric, its angle being fixed. 
Particulars of main valve and its eccentric 
being given, together with the maximum travel 
permissible for the expansion plate : To find 
the respective throws of the expansion 
eccentric for cutting off at given positions of 
the crank or of the piston 61 

Case XXIX. — Variable expansion by altering the throw 
of the eccentric, its angle being fixed. Con- 
ditions as in Case XXVIII : To construct a 
Zeuner diagram, and to delineate the curves 
of port opening (steamway) for the several 
points of cut-off 62 

Chapter IV. — Series A, Expansion Plate. 

Case XXX. — To construct the main and expansion 
valves and eccentrics for a given cylinder face ; 
the lap of main valve, the lead, the point of 
cut-off, the width of the slot, and a position of 
the crank where a certain slot opening is 
desired being given 64 

Case XXXI. — Conditions as in Case XXX : To construct 
the Zeuner diagram, and from it the valve and 
expansion plate, and to delineate the curve of 
port opening (steamway) 67 

Case XXXIL — Conditions as in Case XXX, but with 
double or multiple-slotted expansion plate : To 
construct the valves and eccentrics for a given 
cylinder face, the lap of the main valve, the 
lead, the point of cut-off, the width of the 
slots, and a position of the crank where a total 
slot-opening is desired being given 70 



X. INDEX TO CASES. 



PAGE 



Case XXXIII.— Conditions as in Case XXXII : To con- 
struct the Zeuner diagram, and from it the valve 
and expansion plate, and to delineate the curve 
of port opening (steam way) for diagrams 32, 
32a, and 32b... 75 



Case XXXIV. — Variable expansion by altering the angle 
of the eccentric, its throw being fixed. Par- 
ticulars of main valve and its eccentric being 
given, together with the maximum travel per- 
missible for the expansion plate : To find the 
respective angles of the expansion eccentric for 
cutting off at given positions of the crank or 
of the piston, and to construct the expansion 
plate 81 

Case XXX V>— Variable expansion by altering the angle 
of the eccentric, its throw being fixed. Con- 
ditions as in Case XXXIV : To construct the 
Zeuner diagram, and from it the valve and 
expansion plate ; and to delineate the curves 
of port opening (steam way) for the several 
points of cut-off ... ... ... ... ... 83 



Case XXXVI. — To construct the main and expansion 
valves and eccentrics for a given cylinder face. 
The lap of main valve, the lead, the point of 
cut-off, the width of the slot, and a position of 
the crank where a certain slot opening is 
desired being given 88 



Case XXXVII.— Conditions as in Case XXXVI: To 
construct the Zeuner diagram, and to delineate 
the curve of port opening (steam way) 90 



Case XXXVIII. -Conditions as in Case XXXVI, but 
with double or multiple slotted valve : To 
construct the valves and eccentrics for a given 
cylinder face. The lap of the main valve, the 
lead, the point of cut-off, the width of the 
slots, and a position of the crank where a total 
slot opening is desired being given ... ... 92 



INDEX TO CASES. XI. 

PAGE 

Case XXXIX.— Conditions as in Case XXXVIII: To 
construct the Zeuaer diagram, and from it the 
valve and expansion plate; and to delineate 
the curves of port opening (steamway) for 
diagrams 38, 38a, and 38b 97 

Case XL. - Variable expansion by altering the angle of 
the eccentric, its throw beiog iixed. Particu- 
lars of main valve and its expansion plate 
being given, together with the maximum travel 
permissible for the expansion plate : To find 
the respective angles of the expansion eccentric 
for cutting off at given positians of the crank 
or of the piston ... ... ... ... ... 105 

Case XLI.— Variable expansion by altering the angle 
of the eccentric, its throw being hxed. Con- 
ditions as in Case XL: To construct the 
Zeuner diagram, and to delioeate the curves of 
port opening (steamway) for the several points 
of cut-off 107 

Case XLII. — Variable expansion by altering the throw 
of the eccentric, its angle being fixed. Par- 
ticulars of main valve and its eccentric being 
given, together with the maximum travel per- 
missible for the expansion plate : To find the 
respective throws of the expansion eccentric 
for cutting off at given positions of the crank 
or of the piston 109 

Case XLIII. — Variable expansion by altering the throw 
of the eccentric, its angle being fixed. Con- 
ditions as in Case XLII : To construct the 
Zeuner diagram, and to delineate the curves of 
port opening (steamway) for the several points 
of cut off Ill 

Case XLIV. — Variable expansion by altering the throw 
of the expansion eccentric, its angle being 
fixed : To construct the valves and eccentrics 
for a given cylinder face to fulfil conditions as 
specified, the valve being double or multiple 
slotted, and the expansion plate of the Series 
Ctype 113 



XU. INDEX TO CASES. 

PAGE 

Chapter VI. —Meyer Cut-off Gear. Series D & E. 

Case XLV. — Variable expansion by altering the lap of 
the expansion plate, the throw and angle of the 
expansion eccentric beiag lixed : To construct 
the valves and eccentrics for a given cylinder 
face to fulfil conditions as specified ; the lap of 
the main valve, the lead, the earliest and 
latest points of cut off, and the width of the 
single slot being given 123 

Case XLVI. - Conditions as in Case XLV : To construct 
the Zeuner diagram, and from it the valve and 
expansion plates, and to delineate the curves of 
steam way openings 130 

Case XLVII.— Conditions as in Case XLV, but with 
double or muliiple slotted valves : To con- 
struct the valves and eccentrics for a given 
cylinder face to fulfil conditions as specified ; 
the lap of main valve, the lead, the earliest 
and latest points of cut-off, and the width of 
the double slots being given ... ... ... 137 

Case XLVIII. — Variable expansion by altering the lap 
of the eccentric plate, the throw and angle of 
the expansion eccentric being fixed : To con- 
struct the valves and eccentrics for a given 
cylinder face to fulfil conditions as specified, 
the lap of the maio valve, the lead, the earliest 
and latest points of cut-off, and the width of 
the single slot being given 145 

Case XLIX.- Conditions as in Case XLVIII: To con- 
struct the Zeuner diagram, and from it the 
valve and expansion plates, and to delineate 
the curves of steam way openings 151 

Case L. — Conditions as in Case XLIII, but with double 
or multiple slotted valve : To construct the 
valves and eccentrics for a given cylinder face 
to fulfil conditions as specified ; the lap of the 
main valve, the lead, the earliest and latest 
points of cut-off, and the width of the double 
slots being given ... ... ... ... ... 153 



THE PROPORTIONS AND MOVEMENT OF 
SLIDE VALVES.* 



The invention of the slide valve is usually ascribed to 
Murdoch, the pupil and assistant of Watt, who is said to have 
employed it in an experimental engine about the year 1785. 

There is every reason to believe, however, that its real 
origin is lost in the mists of antiquity, inasmuch as a descrip- 
tion of a "sliding valve" occurs in the treatise entitled 
" SpiritaHa seu Pneumatika," written by Hero of Alexandria, 
who flourished 284-221 b.c. In the fragmentary remains of 
the works of this celebrated philosopher and mathematician 
(collected and published at Paris in the year 1693) are to be 
found numerous devices of the greatest interest to engineers 
of tho present day, including the well-known (Elopile, or re- 
action steam engine, the pneumatic apparatus known as 
Hero's Fountain, a double-acting fire engine, and many others, 
the greater number of which existed probably ages before the 
time of Hero, although the knowledge of them appears to 
have been confined to the priesthood. 

This ancient application of the slide valve may or may not 
have been known to Murdoch. In all probability it was an 
independent invention, so far as he was concerned, derived 
from the four-way cock used by him in a previous engine. 
But in any case the credit of having introduced the slide- 
valve — one of the simplest and most beautiful devices for 
effecting a given purpose which ever entered the mind of any 
engineer — belongs to Murdoch. 

It is hardly possible to over-estimate the influence which 
this simple plate of metal, sliding over three holes in the 
face of the cylinder, has had upon the working of the modern 
steam engine. Suffice it to say that the slide valve is the 
most vital part of its mechanism, and the slightest alteration 
in its proportions or movement is all that is necessary to 
convert the best and most economical steam engine into a 
wasteful and extravagant " coal eater." 

* The Author is indebted to the Proprietors of The Mechanical World, in 
which these articles originally appeared, for their permission to republish 
them in book form. 



2 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

Murdoch has another claim to our gratitude for his inven- 
tion of the eccentric; and, indeed, it v^ould appear that had 
Murdoch's career of invention not been overshadowed — and 
even possibly stifled — by the greater genius of his master, 
Watt, his name, now almost unheard of except by the digger 
and delver into the ancient history of the steam engine, would 
have been enrolled among the benefactors of mankind. 

We have now, however, to deal with the slide valve of the 
present day — the same indeed in principle, and altered but 
little in appearance, but developed, and its capabilities 
extended, to a degree which probably even Murdoch never 
contemplated. 

It is by no means certain to whom we are to attribute the 
invention or introduction of the lap slide valve. The inten- 
tion of the primitive (lapless) valve was undoubtedly to allow 
the admission of steam during the entire stroke of the piston, 
and the merest trifle of lap was given to the valve in order to 
prevent the admission of steam to both sides of the piston at 
once at the moment when the valve was passing over the 
centre of its travel. 

The earliest mention of a lap slide valve which the author 
has observed occurs in the specification of a French paient 
(Hamy's, 1827), closely followed by another (Pequeur, 1829), 
and again by Dusart's, 1832, all of which appear to have had 
the lap equal to the width of the steam port, with the inten- 
tion of working the steam expansively. Almost simulta- 
neously, or between the years 1829 and 1833, lap appears to 
have been in use in some American locomotives. 

A full and very interesting account of an early locomotive, 
constructed in 1836 by Robert Stephenson and Co.,* describes 
very completely the construction and operation of the slide 
valves in use at that period ; and the writer of the description, 
with pardonable pride, alludes to " the great perfection of the 
present locomotives," and asserts that "their superiority to 
the old ones is caused not so much by the application of new 
inventions to them, as by the combination of many former 
ones, and the uniting together of several plans which 
separately would be but of small value." 

After this little flourish of trumpets one almost hesitates to 
take advantage of the writer's candour and reproduce the 
statement that " the negative pressure of the waste steam 
amounts to 30 or 40 per cent, of the positive pressure of the 
steam upon the piston when the engine is running very fast, 
and the power of the engine is diminished nearly one-half." 

* London : John Weale, 1838. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 6 

With this evi«1ence of superiority before ns, it is saddening 
to reflect upon the probable condition of affairs prevailing in 
the cylinders of "the old ones." 

The slide valve of Stephenson's engine had only ^in. of 
lap, and it seems quite clear that its only object was to pre- 
vent both steam ports being opened at the same time. 

The advantages of lead were, however, recognised, as the 
steam port was opened for the next stroke just before the 
piston reached the end of the cylinder, " in order to bring it 
up gradually to a stop, and diminish the violent jerk that is 
caused by its motion being changed." 

It is worthy of remark, iu passing, that in the year 1836 the 
word " travel " was used in a different sense from the present 
accepted meaning of the term; for in order to keep the steam 
on the piston as long as possible the valve moved nearly -|^in. 
beyond the port at each end, or "over-opened," as we should 
say. It is specially mentioned that " this distance that the 
slide moves beyond the port is called the travel." The writer 
of the treatise also mentions, almost apologetically, that "the 
total power is diminished a little on account of the steam 
being shut off before the end of the piston's stroke, the 
extent of this action being, however, very limited, as the 
piston is less than ^in. from the end of its stroke when the 
steam is shut off.'' 

An analysis of the proportions and movement of this valve 
will be found in Section III. of the present series, under the 
heading of " An Old-fashioned Slide Valve." 

It is somewhat strange that the writer of the above des- 
cription, after so minutely detailing the many and obvious 
disadvantages entailed by keeping on the steam during what 
was practically the whole stroke of the piston, should in his 
very next paragraph disclose the remedy, but apparently 
without any real knowledge of the importance of his casual 
remark that "in stationary and marine condensing engines 
the steam has usually very little or no lead, but it is shut off 
at two-thirds or three-quarters of the stroke, giving a great 
amount of expansive action; and the eduction has a great 
deal of lead, the port being nearly full open at the commence- 
ment of each stroke." 

Had these conditions been fulfilled in the locomotive he 
was describing — or, in other words, had lap been added to the 
slide valves, as well as the lead he attaches such importance 
to,— a very much better result would have been attained. 
It is evident from the description that the principal object of 
the designer of the Stephenson locomotive was to get as much 
steam into the cylinder as possible, leaving its egress to take 



4 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

care of itself, with the result, of course, that the cylinder 
became choked or gorged with steam, which at anything 
above a very slow speed had to be actually forced or propelled 
out by the piston. 

It is not at all improbable that the wheels would have 
revolved more quickly had the cylinder cocks been kept fully 
open to relieve the congestion. The consumption of fuel 
must, of course, under these conditions, have been enormous, 
particularly as the gab motionf with which this engine was 
fitted afforded no means of shortening the stroke of the valve 
so as to work with some degree of expansion. However, 
this reference is sufficient to prove that lap was in use for 
stationary or marine engines, if not for locomotives, in the 
year 1836. 

Pambour, in his book, " On the Locomotive Engine," pub- 
lished in that year, describes the precise action of a slide 
valve with both a definite lap and lead ; and locomotives were 
at work so constructed very shortly afterwards. 

In " A Practical Treatise on Railroads and Carriages" (2nd 
edition, 1835) Thomas Tredgold, Civil Engineer, writes these 
words : — " An engine ought to be so made that the communi- 
cation between the cylinder and boiler could be cut off at any 
part of the stroke .... at the pleasure of the atten- 
dant, or according to the stress on the engine, instead of the 
usual method of straightening! the steam passage by that 
bungling contrivance which has very properly been termed 
a throttle valve. It ought to be clearly understood that 
everything which interrupts the passage of the steam to the 
cylinder wastes its power." 

This admirable sentence is probably one of the earliest 
recorded references to the now familiar contrivance known 
as automatic expansion gear. This is clearly indicated by 
the words, or according to the stress on the engine, as distin- 
guished from the alternative at the pleasure of the attendant, 
which points to variable or adjustable expansion gear, as we 
should now call it. 

It is difficult to imagine that these words — which disclose 
such an accurate knowledge of the principles of the expansion 
of steam in a cylinder — were written sixty years ago, and 
occur side-by-side with the following paragraph, which, 
though totally irrelevant to the present subject, we cannot 
refrain from quoting. 

t The combined reversing and expansion gear formed by the Stephenaon- 
Howe link motion was not introduced until 1843. 

t Evidently straitening {i.e., narrowing) is here meant. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 5 

Tredgold says : — " Before dismissing the subject of locomo- 
tive eDgines, it ought to be remarked that they must always 
be objectionable on a railroad for general use where it is 
attempted to give them a considerable degree of speed : for 
many pass-places there cannot be : hence the velocity of the 
greater part of the carriages on a line of road must be limited 
by a slow-travelling one. This would not only be a source of 
inconvenience, but one of danger, from carriages striking 
against one another." f 

Railway travellers of the present day will probably envy 
the excellent Tredgold the complacency with which he re- 
gards the possibilities of inconvenience and even danger 
resulting from " the carriages striking against each other." 

It hardly falls within the intention of the present series to 
describe the slide valve in its elementary aspect. 

In a previous work | the author has attempted the task of 
explaining, as to a person supposed absolutely ignorant of the 
steam engine, the principles and construction of the simple 
slide valve. We have here to deal more particularly with 
those modifications of its form and in its manner of move- 
ment which the modern conditions of high pressure, high 
speed, and the fuller utilisation of the expansive force of 
steam, have shown to be necessary or desirable. 

Being therefore supposed conversant with the usually- 
accepted meanings of the terms lap, lead, and travel, as 
applied to a single slide valve, we may proceed to consider 
the several methods of designing a slide valve and its eccen- 
tric to fulfil given coaditions. 

A well-proportioned modern steam engine with a single 
slide valve correctly designed is by no means an uneconomi- 
cal machine. The faults illustrated by our extracts from the 
description of Stephenson's locomotive have been met and 
remedied by the use of the lap slide valve and by the recog- 
nition of the necessity for providing ample facilities for the 
escape of the spent steam. 

There are one or two points upon which it is very neces- 
sary that a clear understanding should be arrived at before 
entering upon the series of constructional diagrams which 
follow. 

Firstly. — We take no account of the inequality caused by 
the angular movement of the connecting rod, but assume that 
the two strokes of the piston — the instroke and the outstroke 



t Tredgold on " Railroads," 1835, p. 88. 

t " The Portable Engine : Its Construction and Management," Lockwood. 



6 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

— are exactly similar. We shall therefore only concern our- 
selves with what takes place upon one side of the piston, or, 
in other words, assume our engine to be single acting. 

Secondly. — We shall suppose, for the sake of simplicity, the 
" throw," or radius of the crank, to be reduced to that of the 
eccentric actuating the valve, so that the circle representing 
the eccentric's path may also denote the path of the crankpin 
to a reduced scale. It may be remarked in passing that it is 
always desirable in practice to draw these figures to as large 
a scale as conveniently possible, in order to minimise con- 
structional errors. 

Thirdly. — Beal and Assumed Positions of the CranJc. The 
real position of the crank in all the diagrams is to the left 
hand of the reader, and will be always denoted by the letter 
A ; and the angular position of the eccentric, as shown in the 
figures, or found by construction from the given data, is al- 
ways with reference to this real position of the crank at A, 
the angle of the eccentric with the crank being invariably 
denoted by the letters A O C. But for all other purposes of 
reference the crank is to be supposed to start from the point 
B upon the right-hand side of the diagram, or diametrically 
opposite to its real position. 

The leason for this apparent contradiction is that instead 
of revolving the entire diagram, with crank, eccentric, points 
of cut-off, release, etc., upon the fixed horizontal base-line A 
B, it is much more convenient in practice to consider the dia- 
gram as fixed, and the base-line revolving in the opposite 
direction upon it, 

This assumption saves a good deal of trouble in several 
ways, and gives a perfectly truthful representation of the 
successive operations as they occar in the course of the stroke. 
The arrow, m all cases, denotes the real direction of rotation. 

The diagrams are drawn to a uniform scale, the unit of 
which is not any fraction of an inch or of a foot, but one-sixth 
the width of steam port in the cylinder face. 

In practice, the size of the steam port for a given diameter 
of cylinder is dependent upon the speed of piston, the pres- 
sure of steam employed, and other causes with which we are 
not concerned for the purposes of the present investigation ; 
but whatever the width of the steam port may be, as dictated 
by the special circumstances in any given case, one-sixth of 
its width is the unit by which the proportions and movement 
of the slide valve will be measured, and on this assumption 
our diagrams are framed. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 7 

Expressed in this way, the dimensions of a sHde valve 
which will serve our purpose as an example may be given 
thus : — 

Width of steam port in cylinder face . . - 6 units. 

Lap of slide valve .. .. 6 ,, 

Travel = 2 (lap + port opening) =24 ,, 

Lead 1 ,, 

Total length of valve 48 „ 

Width of exhaust port in cylinder face 12 ,, 

,, cavity in valve 24 ,, 

Exhaust lap (none) = , , 

Width of bars between ports in face 6 , , 

These proportions are convenient in use, and suitable for 
the ordinary requirements of an engine having a simple valve 
with a lixed {ie., non-variable) degree of expansion. 

A scale of units will be given beneath each diagram, from 
which measurements in terms of steam -port width can be 
taken, and the base-line of the diagram will be divided into 
eighths, representing fractions of the piston's stroke (starting 
from the assumed position of the crank). 

For the sake of clearness in the diagrams, as far as practic- 
able the given data will be represented by thin lines, and 
results by heavy black lines or dots, the necessary construc- 
tion lines being shown of intermediate thickness. 

Finally. — The proof of the reasoning upon which the 
solutions of the following problems are based will be found in 
the well-known 31st Proposition of the Third Book of Euclid, 
which enunciates : — " In a circle, the angle in a semi-circle is 
a right angle ; but the angle in a segment greater than a 
semi-circle is less than a right-angle ; and the angle in a seg- 
ment less than a semi-circle is greater than a right-angle." 
The reader who now studies the geometry of the slide valve 
for the first time, is recommended to establish in his own 
mind from the beginning, the principles upon which it is 
based. "A system of geometry proceeds from simple, axio- 
matic, and incontrovertible principles to the demonstration of 
new truths ; and from the combination of truths previously 
known, new truths are continually evolved ; and thus a system 
of geometrical science is established by a continued process 
of logical deduction." 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



CHAPTER I. 

Case I. — Lap, lead, and cut off being given : To find the 

THROW AND ANGLE OF THE ECCENTRIC ? 

Upon the base line A B, with centre O, describe the lap 
circle SEP, with radius O P, equal to the amount of lap ; 
and upon the line A B lay off P Q equal to the required lead. 

Draw the lead line C Q perpendicular to A B, and let the 
radius O D be the position of the crank at which it is desired 
the cut-off shall take place, reckoned from B. 




SCALE OF UNITS 
Diagram 1, 

Then the tangent R C, at right angles to the radius O D, 
will cut the lead line C Q at C, and O C will be the radius 
representing the throw and angle of the eccentric with refer- 
ence to the real position of the crank at A ; and the circle 
A D C B may be described, forming, as suggested in the first 
section, not only the travel circle or path of the centre of the 
eccentric, but also the crank circle representing, upon a 
reduced scale, the track of the crankpin. 

The dotted ordinate D S let fall from D indicates at S the 
proportion B S of the piston's stroke B A at which the slide 
valve closes and cuts off the steam. 



Case II. — Lap, lead, and travel being given ; To find the 

ANGLE OF THE ECCENTRIC AND THE POINT OF CUT-OFF ? 

Upon the base line A B, with centre O, describe the lap 
circle S R P, with radius O P, equal to the amount of lap ; 
and upon the line A B lay off P Q equal to the required lead. 
With O A as radius, describe the travel circle A D C B. 



TEE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




SOAL£ OF UNtTS 

Diagram 2. 

Draw the lead line C Q perpendicular to A B ; the point C 
in the travel circle will be the position of the centre of the 
eccentric. Draw R C a tangent to the lap circle, and the 
radius O D, at right angles to R C, will be the position of the 
crank (reckoned from B) at which the cut off will take place. 

The dotted ordinate D S let fall from D indicates at S the 
proportion B S of the piston's stroke B A at which the slide 
valve closes and cuts off the steam. 



Case III. — The throw of the eccentric, the lap, and the 
POINT of cut-off being GIVEN : To find the lead and the 
angle of the eccentric ? 

Upon the base line A B, with centre O, describe the lap 
circle SEP, with radius O P, equal to the amount of lap ; 
and from the same centre O, with radius A, describe the 
travel circle A D C B. 




SCALE OF UNITS 
iDlAGRAM 3 



10 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



From the point of cut-off D draw the chord D E tangential 
to the lap circle, and cutting the travel circle at E. 

From centre B describe a circle tangential to the chord D E. 
The radius of this circle is equal to the amount of lead. 

From the points D and E in the travel circle describe arcs 
cutting each other at C. Draw the radius O C, and the angle 
A O C will be the required angle of the eccentric with the 
crank. 

The dotted ordinate D S let fall from D indicates at S the 
proportion B S of the piston's stroke B A at which the slide 
valve closes and cuts off the steam. 



Case IV. — The throw of the eccentric, the lead, and the 

POINT OF CUT-OFF BEING GIVEN : To FIND THE LAP AND THE 
ANGLE OF THE ECCENTRIC ? 

Upon the base line A B, with centre O and radius O A, 
describe the travel circle A D C B. From the point B in the 
travel circle as centre, describe the lead circle, with radius 
equal to the amount of lead. 




SCALE OF UNITS 
Diagram 4. 



From the point of cat-off D draw the chord D E tangential 
to the lead circle, and from the centre O describe, touching 
the chord D E, the lap circle S R P; and from the points D 
and E in the travel circle describe arcs cutting each other at 
C. Draw the radius O C, and the angle A O C will be the 
required angle of the eccentric with the crank. 

The dotted ordinate D S let fall from D indicates at S the 
proportion B S of the piston's stroke B A at which the slide 
valve closes and cuts off the steam. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



11 



Case V. — The throw and angle of the eccentric, and the 
LEAD being given : To find the lap and the point of 

CUT-OFF ? 

Upon the base line A B, with centre O and radius O A, 
describe the travel circle A D C B. From the point C in the 
travel circle (which is the centre of the eccentric in its given 
angular position) let fall the perpendicular C Q. 

Upon the base line A B, lay off towards O the distance Q P, 
equal to the desired lead, and from the centre 0, with radius 
O P, describe the lap circle S R P. 




SCALE OF UNITS 
Diagram 5. 

From the point C draw C R tangential to the lap circle just 
found ; and at right angles to C R draw the radius O D, which 
will be the position of the crank when steam is cut off. 

The dotted ordinate D S let fall from D indicates at S the 
proportion B S of the piston's stroke B A at which the slide 
valve closes and cuts off the steam. 



Case VI. — The throw of the eccentric, the lead, and a 
position of the crank where the steam port is to be open 

A certain fixed amount, '■'■ being given : To FIND THE LAP, THE 
angle of THE ECCENTRIC, AND THE POINT OF CUT-OFF ? 

Upon the base line A B, with centre O and radius O A, 
describe the travel circle, and let O E represent the position 
of the crank when steam is admitted, and O F its position 
when the steam port is to be open the given amount. 



* Useful for ensuring a suflBcient port opening, for example, when the piston 
is moving at its greatest velocity. 



12 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 
D R 




DIAGRAM 6. 

From the point E as a centre, and with radius equal to the 
given amount of port opening, describe a circle, and from the 
point F in the travel circle draw the line F V touching the 
circle of port opening. Parallel to the tangent F Vdraw E D, 
a chord of the travel circle, and join O D, which will be the 
position of the crank when the steam is cut off. 

From the points D and E in the travel circle describe arcs 
cutting each other at C. Join O C, which will be the throw 
and angular position of the eccentric. 

From the centre O describe the lap circle, touching the 
chord D E at P. OP will be the amount of lap required to 
open the port the given amount when the crank occupies the 
radius O F. 

It will also be open a like amount when the crank is at 
O F', where the line F V again intersects the travel circle. 
The reason of this is, of course, that at O F' the port is open- 
ing, and at O F it is closing. 

Ordinates let fall from any of the points in the travel circle 
upon the base line A B will denote the fraction of the piston's 
stroke (reckoned from B) at which the respective operations 
occur. 



Case VII. — The lap and the lead of the valve, and a 

POSITION OF the crank WHERE THE STEAM PORT IS TO BE OPEN A 
CERTAIN FIXED AMOUNT, BEING GIVEN : To FIND THE THROW AND 
THE ANGLE OF THE ECCENTRIC AND THE POINT OF CUT-OFF ? 

Upon the base Hue A B, with centre O, and radius O P 
equal bo the amount of lap, describe the lap circle, and lay off 
upon the line A B, P Q equal to the given lead. Upon A B, 
at the point Q, erect the lead line Q C as a perpendicular, 



THE PEOPORTIONS AND MOVEMENT OF SLIDE VALVES. 



13 



and let O F represent the position of the crank where the 
port is to be open the given amount. Lay off upon the line 
O F the distance S T exterior to the lap circle equal to the 
given port opening. 




SCALE OF UNITS 

Diagram 7. 

From the point T draw the line T C at right angles to O T, 
cutting the perpendicular Q C at C. Join O C, which will be 
the throw and angular position of the eccentric and a radius 
of the travel circle A F B. From the point in the travel 
circle lay off the arc C F' equal to C F, and join O F'. The 
steam port will be open the given amount when the crank 
occupies either the radius O F' or the given position O F — in 
the latter case when the port is closing, and in the former as 
it is being opened. 

Tangential to the lap circle draw a chord of the travel 
circle D E, at right angles to the radius O C. Join O E 
(which will be the position of the crank when admission takes 
place) and D (which will be its position when the valve 
closes and cuts off the steam). 

Ordinates let fall from any of the points in the travel circle 
upon the base line A B will denote the fraction of the piston's 
stroke (reckoned from B) at which the respective operations 



Case Vin, — The lead, the point of cutoff, and a position 
OF the crank where the steam port is to be open a certain 

FIXED amount, being GIVEN : To FIND THE LAP, THE THROW OF 
THE ECCENTRIC, AND ITS ANGLE WITH THE CRANK ? 

In any straight line A B take a point O and draw a 
perpendicular through it. Lay off upon this perpendicular a 
distance upwards O L, equal to the given lead, and from L 
downwards make L M equal to the given port opening. Draw 
L T and M J parallel to A B. 



14 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



Let D represent the position of the crank (reckoned from 
B) when the steam is to be cut off, and let O F represent the 
crank when the port is to be open the given amount L M. 




O D will intersect L T at a point K. Bisect the angle 
T K D by the line H K. Produce F O to N where it cuts the 
line M J, and lay off N J equal to N O. 

From the points O and J, with any convenient radius, 
describe arcs cutting one another at a point G. Join G N, 
which will intersect H K in a point E. Then O E (aot K E) 
will be a radius of the travel circle. 

Make T E perpendicular to L T, and E T will be the radius 
of the lap circle. 

The point C (the centre of the eccentric) may be found 
either by drawing the tangent R C at right angles to the cut- 
off radius O D, or by erecting the perpendicular lead line Q C 
at a distance P Q (equal to the lead) from the lap circle.* 

The point C being thus found, O C will be the throw of the 
eccentric, and A O C its angle with the crank, and the port 
will be open the given amount L M when the crank occupies 
the given position O F.' 

It will also be open a like amount when the crank is at 
O F', as explained iu cases VI. and VII,, C F' being equal to 
CF. 

Ordinates let fall from any of the points in the travel circle 
upon the base line A B will denote the fraction of the piston's 
stroke (reckoned from B) at which the respective operations 



4* Case I. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



15 



Case IX. — The lead, the maximum opening of the port, 

AND THE POINT OF CUT-OFF BEING GIVEN : To FIND THE LAP, THE 
THROW OF THE ECCENTRIC, AND ITS ANGLE WITH THE CRANK ? 

In any straight line A B take a point O and draw a perpen- 
dicular through it. Lay off upon this perpendicular a dis- 
tance upwards O L equal to the given lead, and from L 
downwards, make L M equal to the given maximum port 
opening. 

Draw L T and M U parallel to A B, and let O D represent 
the position of the crank (reckoned from B) when the steam 
is to be cut off. O D will intersect L T at a point K. Bisect 
the angle T K D by the line H K. In H K find by trial a 
point E, equidistant from the line M U and from the point 
O (not from K). Draw the perpendicular E U. Then O E 
(not K E) will be a radius of the travel circle, and E T will be 
the radius of the lap-circle. 




The point C (the centre of the eccentric) may be found 
either by drawing the tangent II C at right angles to the cut- 
off radius O D, or by erecting the perpendicular lead line Q C 
at a distance P Q (equal to the lead) from the lap circle.* 

The point C being thus found, O C will be the throw of the 
eccentric, and A O C its angle with the crank ; and the maxi- 
mum port opening (which will occur when the crank occupies 
the radius O C) will be equal to the given amount L M. 

Ordinates let fall from any of the points in the travel circle 
upon the base-line A B will denote the fraction of the piston's 
stroke (reckoned from B) at which the respective operations 
occur. 

• Case I. 



16 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



Case X. — Variable cut-off by a simple slide valve with a 

CONSTANT lead : LAP, LEAD, AND THE POINTS OF CUT-OFF BEING 
GIVEN : To FIND THE SEVERAL THROWS AND ANGULAR POSITIONS 
OF THE ADJUSTABLE ECCENTRIC ? 

The method illustrated by Case I. affords a ready means of 
fixing the several throws and angular positions of a variable 
or adjustable eccentric to effect the cut-off at certain given 
positions of the crank. 

The eccentric, it will be understood, has to be so con- 
structed that it is capable of a movement across the crank- 
shaft, with the object of varying its eccentricity, and is 
usually guided by a slot in a plate fixed alongside it, to which 
it can be clamped at any desired position. 

The several points C, C^, C*, etc., denoting positions of the 
centre of the eccentric, are each found separately, exactly as 
in Case I., thus : — 

Upon the base-line A B, with centre and radius O A, des- 
cribe the circle A D C B, representing the path of the crank- 
pin. From the same centre 0, with a radius P equal to the 
given lap, describe the lap circle, and lay off P Q equal to the 
given lead. Draw the lead line C Q perpendicular to A B, 
and let D, D^, D*, etc., be the various positions of the crank 
at which it is desired the cut-off shall take place ; thus for 
example, D^ corresponds with one-eighth of the piston's 
stroke (reckoned from B), D^ with one-fourth, and so on. 




Diagram 10. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 17 

From each of these radii, O D, O D^, O D*, etc., draw the 
tangents E, C, R 5 Cs, R* C^, etc., cutting the lead line C 
Q at C, C5, C*, etc., respectively. Then O C, O C^, O C*, 
etc., will be the angular positions and throws of the eccentric 
for the respective points of cut-off with reference to the real 
position of the crank at A. 

Thus with the eccentric at O C^, the valve would close 
when the crank (assumed to be starting from B) reached D^, 
and so on. The several travel circles corresponding to the 
radii 0^,0 C*, etc., are not drawn in the diagram to avoid 
complication, but are to be understood. 

Were the tangents R C, R,^ C,^, etc., extended to cut the 
maximum travel circle A D C B, and points in the circle 
taken instead of points on the lead line, the cut-off would 
still occur at the correct positions of the crank, but the lead 
would not be constant, wherefore the radius of the eccentric 
must be made adjustable, as explained, so that constant (or 
invariable) lead may be secured ; or, in other words, that 
whatever the point of cut-off, the admission of steam shall 
always occur at the same part of the piston's stroke. 

Beversing — As it is often convenient to be able to alter the 
direction of running, as well as the grade of expansion, it is 
usual to duplicate the points 0,0^, C*, etc., upon the lower 
half of the diagram — as shown by the dotted lines, -so that 
with the eccentric at any one of these latter points the engine 
would revolve in the direction of the dotted arrow with the 
same degree of expansion as at the corresponding point in the 
upper half of the diagram. 

The mechanical details of this arrangement, as well as of 
all the other forms of variable expansion gear in the examples 
following are purposely omitted as outside the intention of 
the present series. 

Note.— By the use of two separate eccentrics with their 
respective throws arranged for the maximum and minimum 
points of cutoff required, and a link motion, the degree of 
expansion can be varied, either by hand when the engine is 
running, or by the action of a governor. Only, in the latter 
case, the short-throw eccentric should not open the port at 
all, its throw being made equal to the lap. It may be placed 
exactly opposite the crank. 

The throws and angles of the two eccentrics would then be 
represented by A C and A P in Diagram X. 



18 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES 



VALVES WITH CENTRAL ADMISSION. 

For some purposes valves formed to receive their steam 
supply iu the ceutral cavity are employed, the exhaust being 
controlled by the outer edges of the valve, exactly contrary 
to the usual arrangement. Fig. 1 shows a common type of 
piston valve having a steam lap equal to the width of the 
port, and without exhaust lap, the outer edges of the valve 
and ports being coincident. 

Case XI. — In a central admission valve : Lap, lead, and 

CUT-OFF being GIVEN : To FIND THE THROW OF THE ECCENTRIC 
AND ITS ANGLE WITH THE CRANK ? 

In this case, on account of the inverted construction of the 
valve, the real and the assumed positions of the crank occupy 
the same radius O A ; but the assumed direction is contrary 
to the real one. Otherwise Diagram XI. is nothing more than 
Diagram I. inverted ; the same letters are employed, and the 
operations are identical in both cases, thus : — 

Upon the base line A B, with centre 0, and radius O P 
equal to the given lap, describe the lap circle S II P, and lay 
off upon the line A B, P Q equal to the required lead. Draw 
the lead line C Q perpendicular to A B, and let the radius 
O D be the position of the crank at which it is desired the 
cut-off shall take place (reckoned not from B, but from A, as 
explained above). 




Diagram 11. 



Then the tangent B C, at right angles to the radius O D, 
will cut the lead line C Q at C, and O C will represent the 
throw and angular position of the eccentric with reference to 



THE PROPOKTIONS AND MOVEMENT OF SLIDE TALVES. 



19 



the position of the real crank at A, the angle of crank and' 
eccentric for rotation in the real direction being the angle. 
AOC. 




•.am 
Inlci 

ScoJt-c/ iJpit^ 
Case XL, FiCx. 1. 



The crank circle A C D B may now be described, and the 
dotted ordinate D S will indicate, at S, the proportion A S of 
the piston's stroke A B at which the piston valve closes and 
cuts off the steam. 

Diagrams II. to V. can, of course, be also made use ofj 
inverted in the same manner, for their respective cases. 



Case XII. — Exhaust : Throw and angle of eccentric 

BEING given : To FIND THE POINTS OF RELEASE AND CLOSURE OF 
THE EXHAUST ? 

Upon the base line A B, with centre O, and with the given 
radius O A (equal to the throw of the eccentric), describe the 
travel circle A C B F. 

From the given point C in the travel circle, representing 
the position of the eccentric, draw the radius O C and produce 
it to F. Draw the diameter G H at right angles to the 
diameter C F, and if the valve has no inside lap, the release 
will take place at G and the closure at H, the crank being 
supposed to start from B. 



20 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



K 


^ 






/ 


/I 


fC 




. 7 


3 


6 


/ 


/ 

3 




1 


p 


^ 


4 


— F" 


r^^ 


8 


4 


8 




s 


iN^C— 




^I2r— 


V- 


— 18 


i 


2^ 



fSCALE Oi 



Diagram 12. 



To determine the portion of the strohe during which the 
exhausting port^ remains wide open. Lay off upon O F the 
distance O V equal to the width of the steam port, and draw 
the chord J K parallel to G H. The port remains wide open 
to the exhaust while the crank revolves through the arc 
JFK. 

From G to J the port is opening and from K to H it is 
closing. 

Ordinates drawn from any of these points to the line A B 
■will denote the fraction of the piston's stroke at which the 
respective operations occur. 



Case XIII. — Exhaust lap : Throw and angle of eccentric 

BEING GIVEN : To FIND THE EXHAUST LAP (iF ANY) REQUIRED TO 
RELEASE THE STEAM AT A GIVEN POSITION OF THE CRANK ? 

Exhaust lap, added to the inside edges of the valve, has the 
effect of reducing the period of exhaust by delaying the 
release and accelerating the closure of the port. 

Upon the base line A B, with centre O, and with the given 
radius O A (equal to the throw of the eccentric), describe the 
travel circle A C B F. 



Not the exhaust port, but the port which for the moment is exhausting. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



21 



From the given point C in the travel circle, representing 
the position of the eccentric, draw the radius O C and 
produce it to F. Draw the diameter G H at right angles to 
the diameter C F. 




Diagram 13. 

Let G' O be the position of the crank (supposed to start 
from B) at which it is desired that release shall occur. Make 
H H' equal to G G', and draw the chord G' H'. 

The exhaust lap required will be equal to O W on the line 
O F, and with this amount added to the inside edges of the 
valve the steam will be released when the crank is at G', and 
the port will be closed when the crank reaches H'. 

The portion of the stroJce during which the port remains 
wide open may be determined exactly as in Case XII., that is, 
by laying off a distance W V, upon the radius O F, equal to 
the width of steam port, and drawing a second chord J' K' 
parallel to G' H', and distant from it the amount W V. 

The port will be wide open to the exhaust, while the crank 
passes through the arc J' F K' as before. 

Exhaust clearance {or exhaust lead) is the converse of 
exhaust lap, and has the effect of causing the port to be 
opened sooner and closed later in the revolution. 

If the position of crank at which release was to occur were, 
for example, at G°, or anywhere earlier in the revolution 
than at th^ diameter G, then exhaust clearance would be 
indicated, the amount of which would be ascertained precisely 
in the same way as the exhaust lap was, viz., by making 



22 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

H H" equal to G G°, and drawing the chord G^ H^ parallel 
to G H. O W° shows the amount which would have to be 
taken off the inside edges of an ordinary valve, the width of 
whose cavity was exactly equal to the distance acrosa the 
bars on the cylinder face. 

Caution. — Exhaust clearance should be very sparingly 
employed, and in the diagram the point of cut-off D, due to 
the given throw and angle of the eccentric, should be 
distinctly shown, or the valve may be inadvertently con- 
structed so that the exhaust opens before the steam has 
ceased to enter the cylinder. 

Note. The point of release cannot be altered by the use of 
exhaust lap or clearance, without correspondingly affecting 
the point of closure. 

Ordinates drawa from any of the points in the circle to the 
line A B will denote the fraction of the piston's stroke at 
which the respective operations occur. 



CHAPTER II.— ANALYSIS. 

The diagrams which we have hitherto had under considera- 
tion are what may be termed constructional or synthetic 
figures. By means of one or other of them, any given cylinder 
face of the usual three ported type can be readily fitted with 
a valve and eccentric suitable for any requirements of lead 
and cut-off, release or compression, within the effective limits 
of a simple slide valve. But the system of diagrams which 
we shall now enter upon is chiefly valuable as a means of 
analysis for tracing the effects of given proportions in the 
valve, and of the travel, continuously throughout the stroke. 
If, as recommended, the diagrams are drawn out upon a large 
scale, say not less than double the actual size, the movements 
of the valve in relation to the crank or piston can be followed 
and criticised with as much certainty as though they were 
exhibited upon a large -sized model of the valve gear, and with 
these great advantages over the model, that faulty proportions 
can be amended, or suggested improvements tried, with a 
merely nominal amount of trouble. 

The method of Zeuner is the one here generally adopted for 
the analytical diagrams, with some few modifications or 
additions which have proved useful in practice. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



23 



Case XIV.- 

BEING GIVEN : 



-Lap of valve, throw and angle of eccentric 

To find the points of ADMISSION AND CUT-OFF : 



and the POSITION OF THE CRANK OR OF THE PISTON, WHEN THE 
PORT IS OPEN A GIVEN AMOUNT ? 

Upon the base line A B, with centre O and radius O A, 
describe the travel circle A D C B, and from the same centre 
O, with radius O P equal to the given lap, describe the lap 
circle. From the given point C in the travel circle draw the 
radius O C representing the given throw and angle of the 
eccentric relative to the real position of the crank at A. The 
crank is, as before, supposed to have its throw reduced to that 
of the eccentric, and to start from the point B, its assumed 
position. Upon the radius O C as diameter, describe the 
circle O RC P (which will be called the steam circle), and 
draw the radii E and O D, cutting the two circles at the 
points where they intersect — viz., P and R. The portion of 
the steam circle (shaded in the diagram) exterior to the lap 
circle represents port opening. 




scALi: or UNITS 
Diagram 14 and 15. 

A single glance at the shaded crescent is sufficient to show 
that the points of opening and closing the steam port occur 
at the intersections of the lap circle and the steam circle, 
corresponding to positions of the crank at E for the opening, 
and at O D for the closing of the port.''' The maximum 
opening is evidently reached when the crank occupies the 
radius O C, and the distance C^ C measured upon that radius 
will, of course, be 6 units — i.e., the full port opening. 

Any other radius of the travel circle may be drawn, crossing 
the shaded crescent wherever necessary to give the prescribed 
opening of the port. For example, we wish to know where 

* Check this by drawing the chord D E, which should form a tangent to the 
lap circle. 



24 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

the piston is when the port is open 5 units (or f ths of its 
v^idth). There are two points in the revolution where the 
port is open any fixed amount (except the maximum at O C), 
and in the present case we find that the radii O 1^ and 
O 31 both cross at points where the breadth of the crescent 
is 5 units, as measured by the dark Hnes 1^ 1» and 3^ 3^. 
Therefore the port is open 5 units when the crank occupies 
either of the radii O^ or O 3^ —in the latter case when the 
port is closing, and in the former as it is being opened. 
Ordinates let fall from tbe points U and 3^ in the travel 
circle cut the base line A B near the points of ^th and fths 
respectively of the piston's stroke. The port is therefore 
open the given amount of 5 units at ^th, and again at fths of 
the stroke. 

Case XV. — Lap, throw, and angle of eccentric being- 
given : To find the amount the steam port is open for a 
given position of the piston or of the crank ? 

Lay off, upon the base line A B in Diagram 14, the 
proportion of the piston's stroke, reckoned from B, at which 
it is desired to ascertain the port opening. For example, let 
it be required to know the amount the port is open when the 
piston has reached fths of its stroke : Upon the base line 
A B erect a perpendicular at the point fths, cutting the trave 
circle (which is also the crank circle) at 3^. Draw the radius 
3^, representing the position of the crank when the piston 
is at fths of its stroke. The port is at once seen to be open, 
upon this radius, the distance 3^ 3^, which measured by the 
scale of units, is seen to be equal to five units (iths of the full 
port width) ; or, if the diagram be drawn full size, the 
<iistance 3'^ 3^ would give the actual amount of opening, and 
could be measured with an ordinary rule. 

Case XVI. — Lap, throw, and angle of eccentric being 
GIVEN : To find all the points in the cycle of operations 

FOR ONE revolution OF THE CRANK, THE SLIDE VALVE HAVING 
NEITHER EXHAUST LAP NOR CLEARANCE ? 

Diagram 16, which contains all that is necessary to examine 
the operations of the slide valve during one revolution of the 
crank, is, so far as the steam portion of the cycle is concerned, 
a reproduction of Diagram 14, but with the addition of an 
exhaust circle and the radii connected with it. Upon the 
base line A B, with centre O and radius O A, describe the 
travel circle A C B F (which is also the crank circle). The 
given point C in the travel circle represents, as before, the 
position of tbe centre of the eccentric with reference to the 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



25 



real position of the crank at A, and the angle A O C is the 
angle between them. From the point C draw the diameter 
C F, and, at right angles to C F, the diameter G H. From 
the centre O, with radius O P equal to the given lap, describe 
the lap semt-circle L E, P M. Again, from the same centre O, 
with radius O S equal to the width of steam port, describe 
the semi circle L S T M. In the present case, the width of 
port being equal to the lap, the two semi-circles are of the 
same radius and appear as a continuous circle ; but it must 
not be lost sight of that the semi-circle upon the exhaust half 
of the diagram -^.e., below the diameter G H of the travel 
circle — has its radius equal to the width of steam port, when 
there is no exhaust lap (see Diagram 17)- 

Upon the radius O C as a diameter describe the steam 
circle O R C P, and upon the radius O F as a diameter 
describe the exhaust circle O T F S. Draw in the radii O E, 
O D, O J, O K, cutting the intersections of the several circles 
at P, R, S, and T, respectively, and the Zeuner diagram is 
complete. 




ICAl,/-: OF UNITS 
Diagram 16. 



We can now proceed to review the cycle of operations due 
to the given proportions of the valve and eccentric, starting 
from the point B, the assumed position of the crank. We 



26 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



find upon the radius O B the port open a distance P Q, which 
is the lead of the valve. The port attains its maximum 
opening when the crank arrives at C, at which point the 
valve, having reached the extremity of its travel, begins to 
return, and the cut- off occurs when the crank reaches D. 
The arc E C D marks the period of admission. From D to 
G is the period of expansion. The supply of steam has been 
cut off, but the elasticity of the imprisoned steam continues 
to impel the piston until the crank reaches G, where the port 
is opened to the exhaust and the expanded steam is released. 
This opening of the port to the exhaust is indicated by the 
radius O G forming a tangent to the exhaust circle O T F S. 
The shaded lens-shaped figure O T V S denotes the port 
opening for the exhaust, and it will be seen that when the 
crank is at O J the port is fully open, the dark Hue O S 




.«_ 6 __ /2 _ IS - U- JO - 36 — ->« — • (s 

Scale, of Unila 



Case 16. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 27 

measuriDg by scale six units, or full port width. The port 
remains wide open until the crank reaches K, when closing 
commences ; and the radius O H Rgain forming a tangent to 
the exhaust circle, indicates the clo&ure of the port. From 
G to H is the period of exhaust. 

Referring to the three small sketches of the slide valve and 
cylinder face, it will be seen that while the crank was passing 
from D to G the valve moved from the position shown in 
Fig. 1 to that indicated by Fig. 2, the latter showing the 
inner edge of the valve just beginning to uoclose the steam 
port to the right. 

When the crank is at F (see Fig. 3) it will be observed that 
the valve, now at the extremity of its travel, is much farther 
to the right than is necessary for the full opening of the port 
to the exhaust. 

This length of travel is for the purpose of allowing the left- 
hand port in Fig. 3 to be fully uncovered. 

We have no concern with this other end of the valve at 
present, however, but merely introduce this third figure to 
account for the circumstance that theexhaustiog port remains 
wide open (with this proportion of valve) for about one-third 
of the revolution, only beginning to close when the crank 
reaches K. 

The reason why the exhaust port is made double the width 
of the steam ports is also evident when the position of valve 
in Fig. 3 is observed — half covering the exhaust port, but still 
leaving an opening the full width of the steam port. 

To continue the cycle : From H to E is the period of 
compression, during which the steam at atmospheric pressure 
which has remained in the cylinder is, so to speak, heaped up 
by the advancing piston, preparing the way for the incoming 
rush of live steam, which will take place when the port opens 
at E, thereby greatly reducing the shock upon the piston. 
From E to B is the period of lead or preadmission. 

The compression and the lead together play an important 
part in assisting the easy reversal ( f the motion of the piston 
by interposing an elastic cushion or buffer through which the 
strains upon the framing of the engine are greatly reduced. 
W<-! have now reached the point B. from which we started in 
the revolution, and have ascertained the point at which every 
operation in the cycle takes place. These points in the cir- 
cumference of the travel circle can, of course, be transferred 
by ordinates to the base line A B, as before, and the fractions 
of the piston's stroke noted for all the operations. 



28 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES 

Case XVII. — Lap, throw, and angle of eccentric being 
GIVEN : To find all the points in the cycle of operations 

FOR one revolution OF THE CRANK, A GIVEN AMOUNT OF EX- 
HAUST LAP BEING ADDED TO THE INNER EDGES OF THE SLIDE 
VALVE ? 

Diagram 17 is a facsimile of Diagram 16. but with the 
necessary additions for an exhaust lap of one unit. After 
proceeding as in Case XVI., as far as the steam side of the 
diagram is concerned, and having drawn the diameter G H at 
right angles to the diameter C F, complete the diagram thus: — 

From the centre O, with radius O W equal to the given 
amount of exhaust lap, describe the exhaust lap semi-circle, 
and with the same radius describe a semi-circle at each of the 
points G and H as centres ; and from the centre O, with 
radius O V equal to the width of steam port plus the amount 
of exhaust lap, describe the arc S' V T' within the exhaust 
circle. 




SCALE OF UNITS 

Diagram 17. 



At the intersections S' and T' respectively, draw in the 
radii O J' and O K', and tangential to the two semi-circles at 
G and H draw the radii O G' and O H' respectively, which 
will cut the intersections of the exhaust circle O S' F T' and 
the exhaust lap semi-circle. (As this latter semi-circle is 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 29 

usually so small that it is dijfficult to locate the exact points 
of intersection, the use of the small semi-circles at the cir- 
cumference, as here given, is desirable for the sake of 
accuracy.) 

The effect of the addition of one unit of exhaust lap will 
now be seen on the diagram. The release occurs at G', 
instead of at G, and the full opening of the port at J' instead 
of at J — both later in the revolution. The end of the period 
of full port opening occurs at K' instead of at K, and the 
closure at H' instead of at H — both earlier in the revolution, 
thus shortening the period of exhaust at both ends, giving a 
longer period of expansion, and a longer compression period. 

This is all that is really necessary to be done in delineating 
the exhaust period in a diagram of this kind ; but there is a 
small correction which, though perhaps not of great import- 
ance in itself, is nevertheless worth brief consideration. 

Referring again to Fig. 3, Case XVI., it has been before 
noticed that at the extreme position of the valve the exhaust 
port is partially covered by the inner edge of the opposite 
end of the valve. When there is no exhaust lap, the obstruc- 
tion amounts to one-half the port, for which reason, among 
others, the exhaust port is made of a width equal to two 
steam ports. 

But with the addition of exhaust lap to the valve this 
obstruction amounts to half the port, p^us the exhaust lap, 
and therefore the available exhaust port opening is somewhat 
reduced as the valve approaches, and recedes from, the ex- 
tremity of its travel ; the maximum obstruction on the radius 
O F being equal to the exhaust lap (shown dotted in Fig. 3, 
Case XVI.). 

The correction is made thus : —On Diagram 17 produce the 
radius O F to Z or further, and upon this radius from the 
centre O lay off O V equal to the width of the steam port. 
Now, with radius O A (equal to that of the travel circle) and 
centre Z, describe the arc aYb passing through the point V 
just found. 

Through the intersections a and h draw the radii O X and 
O Y, and the shaded figure within the exhaust circle will 
represent the correct port opening to the exhaust at any point 
where a radius representing the crank may be drawn. 

Ordinates from any desired radius of the travel circle may 
be drawn to the base-line A B, giving the position of the 
piston at the various points in the cycle as before. 



30 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES 



Case XVIII. — Variable cut-off by a simple slide valve 

WITH A CONSTANT LEAD. LaP LEAD, AND THE POINTS OF CUT- 
OFF being given, AND THE SEVERAL THROWS AND ANGULAR 
POSITIONS OF THE ADJUSTABLE ECCENTRIC BEING LAID DOWN AS 

IN Case X. : To construct the Zeuner diagram there 

BEING NEITHER EXHAUST LAP NOR CLEARANCE.*, ' ,jj 

Having laid down the throws and angular positions of tHe 
adjustable eccentric by Case X., proceed to describe upon^the 
radii O C, O C"^, O C*, etc. (representing the throws of the 
eccentric) as diameters, the steam circles O E, C Q, etc., down 
to O R^ C^ Q, the smallest. (For the sake of clearness in the 
diagram, the intermediate circles R^ C'^ Q, O E* C*^Q, etc 
are omitted.) 




SCALE or UJ^JT^ 
Diagram 18. 



Draw in, at the intersections of the several steam circles 
with the lap circle, the radii O E, O Ei, and the intermediate 
radii O E^^, O E*, etc. (omitted in the diagram) representing 
the positions of the crank at the respective points of admission. 

From these we see that though the lead is constant {i.e., 
the steam port is always open the same amount, P Q; when 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 31 

the crank is at B, the assumed starting point), yet admission 
begins earher with each ^ucces^^ive earlier point of cut-off, 
until, if the eccentric were set opposite to the crank at Q, the 
periods of admission and of pre-admission would be exactly 
equal. 

Draw in also the radii D, O D', etc., through the inter- 
sections E, R', etc , of the steam circles with the lap circle 
representing the positions of the crank at the successive 
points of cut-off. 

From the centre O, with the successive radii C, O C, 
etc., describe the several travel circles (of which only two — 
the full travel, and that for the ^th cut-off — are shown), and 
the diagram is complete so far as the steam side is concerned. 
As it would needlessly complicate the figure to show all the 
exhaust circles, we shall confine ourselves to the two corres- 
ponding to the maximum and minimum points of cut-off. 

Therefore, produce the radii O C and O C to F and F', and 
upon O F and O F', as diameters, describe the respective 
exhaust circles. 

Draw in the intersecting radii O J and O K, O J' and O K', 
and the diameters G H and G' H' of the travel circles at right 
angles to C F and C F' respectively. For convenience of 
examination it is as well to shade the steam crescent R C Q 
and the corresponding lens-shaped figure O S T on the 
exhaust side of the diagram, and to shade or darken much 
more deeply the corresponding figures which deHneate the 
steam and exhaust openings for the minimum cut-off. 

In the diagram before us the width of steam port is equal 
to the lap ; but where this is not the case, the semi circle 
upon the exhaust half of the diagram (supposed to be divided 
by the diameter G H for the maximum cut-off and by the 
diameter G' H' for the minimum) must be made, as previously 
mentioned, of a radius equal to the width of steam port. 

Finally, the radii O C^ to C^ should be produced to the 
travel circle if it be desired to note the positions of the crank 
at the respective points of fullest port opening; and ordinates 
may be drawn to the base line A B to indicate the fraction of 
the piston's stroke reckoned from B, at which any or all of 
the operations occur. 

The cycle of operations for any or of all the points of cut-off 
may be followed exactly as explained in Case XVI. ; the same 
reference letters (distinguished by the small index figures) 
serving for the different cycles shown on the diagram. 

Beversing. — No attempt is made to show the duplicate 
positions of the eccentric for reversing, upon the present 
diagrtm, but it is to be understood that if the crank be 



32 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

required to revolve in the direction of the dotted arrow, all 
the points C, C^, C*, etc. on the lead line C Q would be trans- 
ferred to corresponding positions helow the base-line A B. 



Case XIX. — Valves with central admission (as piston 
VALVES — see Case XI.). Lap, throw and angle of eccentric 

BEING given : To FIND ALL THE POINTS IN THE CYCLE OF OPERA- 
TIONS FOR ONE REVOLUTION OF THE CRANK ? 

Upon the base-line A B, with centre O, and radius O A, 
describe the travel circle A F B C (which is also the crank 
circle), and from the same centre O, with radius O P, equal 
to the given lap, describe the lap semi-circle L R P M. 

Again, from the same centre O, with radius O S equal to 
the width of the port, describe the semi-circle M T S L. 

The given point C in the travel-circle represents, as before, 
the position of the centre of the eccentric with reference to 
the real position of the crank at A, and the angle between 
them is the angle A O C. From the point C draw the 
diameter C F, and at right angles to F the diameter G H. 

Upon the radius O C, as a diameter, describe the steam 
circle O R C P, and upon the radius O F as a diameter de- 
scribe the exhaust circle O T F S. Draw in the radii O E, 
O D, O J, O K, cutting the intersections of the several circles 
at P, R, S and T respectively?, and tbe diagram is complete. 




Diagram 19. 



THE PROrORTIONS AND MOVEMENT OF SLIDE VALVES. 



sa 



Starting from the point A, which with this inverted con- 
struction of valve is the assumed as well as the real position 
of the crank, we find, upon the radius O A, the port open a- 
distance Q P, which is the lead of the valve. Moving down- 
wards in the direction of the dotted arrow, we find the port 
fully open at C, the valve having reached the extremity ot 
its travel. The cut-off occurs when the crank lies upon the' 
radius O D, which completes the period of admission. Fronoi 
D to G is the expansion period, and at G (there being nO 
exhaust lap specified) the exhaust period begins. Port open- 
ing to the exhaust is shown by the shaded lens-shaped figure 
O T VS. 

The port is fully open for exhausting (the distance O S) 
when the crank reaches O J, and remains wide open 
until the crank arrives at K, where it begins closing, and is 
quite closed at H. From H to E is the period of compression, 
and fresh steam is admitted at E in readiness for a new 
stroke. E A is called the period of preadmission or lead. 

There is one point in connection with valves of this kind 
which should be noticed before going further. Fig. 1 shows 
the same valve as Fig. 1, Case XI., but as having moved to 



m w/^w/A 1 


W/M. 








1 






'/A v7//A v7/A 


m 


i 



6 — 12 — 18— U — A)— 36— 
ScaXt of UnUi 

Fig. 1— Case 19. 



one extremity of its travel. It will be seen that in the 
position the valve now is, the lap half obstructs the steam 
inlet. As the inlet is twice the width of the steam port (just 
as the ordinary central exhaust port is), this still leaves a 
clear space equal to the steam port for the passage of the 
steam. It will also be observed that the bars separating the 



§4 THE PEOPORTIONS AND MOVEMENT OF SLIDE VALVES. 

ports are wider than usual. This addition to their vs^idth is 
equal to the lap of the valve, and has the effect of removing 
the effective lap of a given valve from its outer to its inner 
edges. 

Were the bars of the ordinary width, which may be ex- 
pressed as half the exhaust cavity, minus half the exhaust 
port, the steam and exhaust periods on the diagrams would 
exchange places. 

Convenient proportions for small valves of this kind would 
be as follows (width of steam ports being taken as 6 units) : — 

Inside (steam) lap of valve 6 units. 

Travel 24 

Lead , 1 

Exhaust lap 

Width of central steam inlet 12 

Width of cavity in valve 24 

Total length of valve 48 



Case XX. — An old fashioned slide valve. 

In " A Description of the Patent Locomotive Steam Engine 
of Messrs. Kober^./ Stephenson and Co., Newcastle-upon- 
Tyne," the slide valve used for a locomotive built for Messrs. 
Cubitt in 1836 is minutely described, and it will be instruc- 
tive, and perhaps not uninteresting, for us to put it into the 
form of a Zeuner diagram, analysing its movements in 
relation to the piston, with a view of noting its action and 
comparing its proportions with those of the valves we have 
had under consideration. 

Its performance can hardly be considered satisfactory, as 
" the back pressure amounted to 30 or 40 percent, of the 
positive pressure of the steam upon the piston," to quote 
from tDe detailed description. 

The dimensions were as follows : — 

Inches. Units. 

Width of steam port ^ 1 .... 6 

,, exhaust port ^.. 1^ .... 9 

,, bars 1 6 

Between extreme edges of steam ports 5^ 33 

Length of valve 5t .... 33| 

Travel 3 .... 18 

Lap 0^ .... 01 

Exhaust lap Ofg Of 

Length of cavity in valve 3| . . _ 20| 

Lead : Steam comes on ^in. before piston reaches end of its 

stroke. 
Cut-off : About gin. before end of stroke. 
Diameter of cylinder, 12in. 
Stroke „ ISin. 

Working pressure in boiler, 50lb. per square inch. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



85 



To Construct the Diagram from the Given Dimensions — 
Draw the base-line A B, and with centre O and radius O A 
= l^in. describe the full-size travel circle - the same circle 
answering for the crank circle upon a scale of one-sixth, the 
actual stroke of piston being 18in. 




Diagram 20. 



Lay off, from B, the distance B I equal to ^^^in. {i.e., ^m. 
upon a scale of ^th), and let fall the ordinate Z E. E repre- 
sents the point of admission. From the centre O describe 
the lap circle ^in. radius and from E draw E D tangential to 
the lap circle. 

D is the point of cut-off, and an ordinate D c let fall from 
D represents the position of the piston when cut-off occurs. 

Draw in the radii O E and O D, and at right angles to D E 
draw the diameter C F. Upon the radii O C and O F as 
diameters, describe the steam and the exhaust circles respec- 
tively ; O C represents the throw of the eccentric, and its 
angle A with reference to the real position of the crank 
at A. 






36 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

From the centre O, with radius of l^in. equal to width of 
port phis lap, describe the arcs V U and S T. The figures O 
V U and O S T represent port openings for steam and for 
exhaust respectively. 

Draw the diameter G H at right angles to C F, and at G 
and H, with radius of ^in. (equal to the lap for both steam 
and exhaust), describe the small circles G' and H'. Draw in 
the radii O G' and O H', and the diagram is complete. 

Starting with the crank at the assumed position B, we find 
the steam port open a distance P Q of about ^in., showing 
rather considerable amount of lead. At C^ tbe port is fully 
open, and remains so until C^ is reached, the valve having 
moved out to the extremity of its stroke at C, just as in the 
exhaust circles in Cases XVI. and XVII. As the diagram is 
actual size, all these port openings can be measured directly 
upon their radii. 

It will be observed, as noticed in Section I., that the valve 
moves about ^in. more at each end of its stroke (the dis 
tances V C and S' F respectively) than is necessary to fully 
open the port. 

Continuing the cycle, the steam is cut off when the crank 
reaches the point D in the travel circle corresponding with 
the point c in the piston's stroke, and at G' the exhaust 
opens. The exhaust cycle is exactly the same as that of the 
steam, and differs in only one respect from the exhaust cycle 
in Case XVII. — viz., in the exhaust lap being one-sixteenth 
of the width of port instead of one-sixth. 

Owing to the steam being kept on during almost the entire 
stroke, the back pressure must have been enormous, the 
cylinder being choked with steam, which could not get away. 
An outside lap of half-an-inch, with a corresponding advance 
in the angle of the eccentric, would have materially improved 
the performance of the engine. 



Case XXI. — A suggested improvement in the delineation 
OF the exhaust port opening in the Zeuner diagram. 

Diagram XXI. shows the usual lens-shaped figure O T V S 
representing the port opening for the exhaust darkly shaded ; 
and a much larger figure shaded more lightly, bounded by 
J L M K 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVE?. 37 




SCALE. OF UNITS 
; Diagram 21. 



A closer examination will show that both these figures are 
contained between the same radii, and are both of the same 
radial breadth. The radii O G and O H mark the points of 
opening and closing respectively in both cases, and each of 
the figures attains its full breadth upon the radii O J and 
OK. As a matter of fact the larger figure is simply the 
transference or projection of the smaller one to the circum- 
ference of the travel circle, where it delineates the exhaust 
opening at the same distance from the centre of the diagram, 
and consequently upon the same scale as the steam-port 
opening. 

A much juster idea of the real proportions existing between 
the port openings for steam and exhaust is derived from this 
projection than is the case with the usual method, and the 
importance of a full and free exhaust is so obvious to every 
designer, that no apology is needed in introducing this 
departure from established practice. 

With the proportions we have been using, viz., lap equal to 
width of steam port, the bounding curves J L and M K are 
very nearly approximated to by describing arcs from a centre 
X, found by trial, equi-distant from the four points J, L, M, K. 



38 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

The similarity in length between the two figures upon the 
radii lying between O G and O J is evident at once. For 
example, upon N the distance O o is almost, but not quite, 
equal to the distance m n. 

Where the port opening and lap are not equal, tho port 
opening is measured from the circumference towards the 
centre, and the bounding curves are found by transferring 
the distance upon each radius from the inner, or usual figure, 
to the outer one by actual measurement, the lengths thus 
found giving points in the curve. 



CHAPTER III. 
THE DOUBLE SLIDE YALVE— Series A. 

In all steam engines making any pretence to economical 
working some arrangement has bo be provided for securing an 
early point of cut-off, with, as a result, a proportionately 
greater degree of expansion in the steam admitted to the 
cylinder. 

Although a simple slide valve may, by means of an adjust- 
able or movable eccentric, be used within certain limits as a 
variable expansion gear, yet owing to the circumstance that 
the freedom of the exhaust is much interfered with at the 
earlier points of cut-off, it becomes desirable, when a greater 
.degree of expansion than about one-half is required, to make 
use of a separate valve to effect an early closing of the steam 
port, in order that the valve which controls the exhaust may 
retain its full travel under all circumstances. 

The contrivances extant for producing this result are end- 
less, and embrace every description of slide valve — single, 
double, and even triple, actuated by eccentrics, cams, levers, 
or other mechanical devices for producing reciprocating 
motion, and in the majority of cases variable as to the degree 
of expansion afforded, either by hand or automatically by the 
action of a suitable governor. 

In addition to these, there is a large and varied class of 
lifting or poppet valves, usually of the double-beat or Cornish 
type, equilibrated and so arranged as to be tripped or released 
at a point in the stroke determined by the governor. 

There are also two disguised forms of slide valve, well 
known under the names of Corliss and piston valves 
respectively. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 39 

The latter of these is in principle simply a common slide 
valve bent into a circle round its spindle as an axis, and 
reciprocated in the usiial manner/'^ The first named is also a 
slide valve bent into a circular form, but at right angles to 
the direction of motion ; and its spindle or axis does not 
slide, but is partially rotated backwards and forwards within 
a circular bored- out seat by means of a rocking lever, just as 
an ordinary plug-tap is opened and shut by a lever handle. 
The Corliss valve, however, is widely different in principle 
from the tap or cock which it so much resembles, being a 
true slide valve, though working upon a curved face instead 
of a flat one. 

These have all grown out of the necessity for a valve which 
should give an early and sharp cut-off as regards the steam, 
without at the same time restricting the outlet by which the 
spent steam escapes into the atmosphere or into the 
condenser. 

It is obvious that if the admission and the exhaust are to 
be independent of one another, two separate valves are 
necessary, whereby that governing the exit of the steam is 
unaffected by any variation in the travel or relative position 
of the admission valve. 

These two valves may be placed each in a separate steam 
chest; but the usual and more convenient arrangement is 
that of a second or cut-off valve, working upon the back of 
the main valve, and controlling the inlet of steam through 
ports formed for the purpose through the latter. 

Even of this arrangement there are many varieties, and we 
shall proceed to consider a few of the more useful variations 
of this type. 

For the sake of clearness and brevity in description we shall 
speak of the cut-off valve as the expansion plate, or ihe plate 
simply ; and the term valve will be understood to refer 
exclusively to the main slide valve itself. Further, in order 
to distinguish the ports in the main valve from those in the 
cylinder face, we shall invariably speak of the former as slots. 
Thus the valve opens and closes ports in the face of the 
cylinder, and the plate performs the same functions with 
regard to the slots in the valve. 

It may be as well to mention also that all the diagrams 
are constructed for a rotation in the direction of the hands of 
a watch, as denoted by the arrow in each case, the assurned 
direction being, as before, opposite to the actual one. 

* See Cases XI. and XIX., Chapters I. and II. 



40 THE PROPOKTIONS AND MOVEMENT OF SLIDE VALVES. 

For an anti clockwise rotation the diagrams would be 
reversed, as though seen from the back of a tracing or in a 
looking-glass. 




Scalcof (/nitf 
Fig. 1, 



-5^ €6 



Fig 1 represents an ordinary slide valve, with the necessary 
addition to its length at each end and the slots through 
which the steam passes into the ports in the cylinder face. 
The cross-hatched portion represents the original valve, and 
it will be seen that the distance apart of the slots cut through 
the valve is equal to the length of a simple valve having a lap 
somewhat less than six units. 

The expansion plate is a simple flat plate whose length is 
just sufficient to close both slots when placed in its middle 
position upon the valve. An expansion plate if longer than 
this would be said to have positive lap ; if shorter, negative 
lap. This particular form of plate is correctly described as 
having neither positive nor negative lap, and will be known 
for the sake of distinction as Series A. It must be borne in 
mind that in a combination such as we are describing, the 
functions of the main valve differ importantly from its duties 
when used as a simple slide valve, as in the former case the 
cut-off is performed by the expansion plate alone, leaving the 
main valve to deal only with the pre-admission (lead) and 
exhaust opening and closure. 

For this reason it is usual to reduce the lap of the main 
valve somewhat, with the object of correspondingly reducing 
the travel. So far as the point of cut-off of the main valve 
is concerned, the only conditions are that it must close not 
earlier than the maximum cut-off of the expansiun plate, and 
not so late as to be still open when the latter re-opens a slot 
later on in the stroke. 

These conditions are, however, very easily complied with, 
and any error in this respect is visible at once upon the 



THE PROPORTIONS AND MOVEMENT OP SLIDE VALVES. 



41 



diagrams by the method we propose to employ for the purpose 
of delineating the port opening in the several combinations of 
two slide valves which we shall now proceed to examine. 



Series A. — Expansion Plate to Cut Off with the Outer 
Edges, and having neither Positive nor Negative Lap. 

Case XXII. -To construct the main and expansion 
eccentrics for a given cylinder face ; THE lap of main 

valve, THE lead, THK POINT OF CUT-OFF, THE WIDTH OF THE 
SLOT, AND A POSITION OF THE CRANK WHERE a CERTAIN SLOT 
OPENING IS DESIRED, BEING GIVEN. 

Upon the base-line A B, with centre O and radius O P equal 
to the amount of lap, describe the lap circle, and lay off upon 
the line A B, P Q, equal to the given lead.* Upon A B, at 
the point Q, erect the lead line Q C as a perpendicular, and 
let O 3 represent the position of the crank (supposed to start 
from B) where the port (as well as the slot) is to be open the 
given amount (hereafter known as the steam way). 




T5' 



18^ 



SCALE OF US ITS 

Diagram 22. 



Lay off upon the line O 3 the distance S T exterior to the 
lap circle, equal to the given port opening. 



* We may define lead for present purposes as the amount the port is open 
at the beginning of the piston's stroke. Preadmission is the distance, or 
fraction of the stroke, left uncompleted at the moment the port begins to 
open. 



42 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

From the point T draw the line T C at right angles to T, 
cutting the perpendicular Q C at C. Join O C, which will be 
the throw of the main eccentric, and the angle A O C will be 
its angle with respect to the real position of the crank at A 
(Oase VII.). 

Let O 4^ represent the position of the crank where the 
expansion plate is to cut off the steam. Draw a? at right 
angles to O 4^, and upon O 3 lay off a distance O z equal to 
the given slot opening when the crank is at O 3. Draw ^ a? at 
right angles to O 3, cutting O a? at a;. Ox would be the 
throw, and A O a; the angle with respect to the real position 
of the crank at A, of an eccentric which would fulfil the given 
conditions as to cut-off and slot opening, if the main slide 
valve were fixed, forming a stationary face for the expansion 
plate to work upon. 

But in the present series the expansion plate is required to 
effect the prescribed conditions as to cut-off and slot opening 
upon a moving main valve driven by an eccentric, whose 
throw and angular position with respect to the crank are 
denoted by A O C. 

All that is necessary to accomplish this movement is to 
refer the line O a? to the point C as an origin. Thus C X is 
to be made parallel with and equal to O x. 

The point X is the centre of the eccentric for moving the 
expansion plate upon the back of the main valve (whose 
eccentric centre is C) to fulfil the given conditions ; and join- 
ing O X, we have A O X as its angular position with regard 
to the real position of the crank at A. 

This can be done in an even simpler manner by construc- 
tion direct from the centre of the main eccentric C thus (see 
dotted lines on Diagram XXII.) : — 

Having fixed the point C so as to fulfil the prescribed con- 
ditions, as in the earlier part of the present case, or otherwise, 
draw the line Z C at right angles to O 4|, and produce it to X, 
or further. From C as a centre, and with radius equal to 
the desired slot opening when the crank is upon 3, describe 
a short arc W Y, and touching this arc draw the line W X at 
right angles to 3 produced. W X will cut Z X at X, which 
will be the centre of the expansion eccentric, as before. 

For the sake of distinction, we shall call the line drawn at 
right angles to a crank position, and passing through the 
point C to X (as Z C in the present case), the director. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



43 



Case XXIIL— Conditions as in Case XXII. To construct 

THE ZeUNER diagram AND TO DELINEATE THE CURVE OF PORT 

opening! (steamway), 

In the preceding Case XXII. we have found by construction 
the angular positions and throws of the main and expansion 
eccentrics for a given point of cut-off with a prescribed 
amount of steamway at an earher position of the crank or of 
the piston. It now remains for us to prove by the application 
of a diagram upon a principle based upon that of Zeuner, that 
the required conditions have been really fulfilled. In Diagram 
XXIII., now before us, the throw and angle A C of the 
main eccentric, and the throw and angle A O a; of the 




SCALC Of vy/Ts 



Diagram 23, 



supposititious eccentric for moving the expansion plate upon 
the back of a fixed| main valve, are those found by Case 
XXII. The lap of the main valve is 4 units, the lead is 1 
unit, the point of cut-off is ^ of the piston's stroke, reckoned 
from the assumed starting point B (or, as the stroke is divided 
into eighths, at 4^ eighths of the stroke). The width of port 
is 6 units, and the width of the slot in the main valve is 6 
units also ; and, j&nally, the steamway, or the clear opening 
through both slot and port, is to be 5 units at f of the stroke. 

+ Really the curve of port clonn-, but the term is to be understood as 
meaning the opening or steam passage due to the combined action of the 
main slide valve and the expansion plate. 

i It is convenient, when investigating the movement of an expansion plate 
upon the back of a moving main valve, to consider the latter as a fixed face 
and the expansion plate as driven by the imaginary eccentric A O a;. Then, as 
already explained, by making C X equal to. and parallel with, O x, the real 
eccentric A O X will give the same relative movement to the expansion plate 
when the main valve is moved by the eccentric A O C. 



44 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

These are the given conditions, and the diagram will soon 
show whether we have adopted the correct proportions. 

First of all, with centre O and radius O C describe the main 
travel circle A D E, and upon O C as a diameter describe 
the main steam circle O K C P, and again from centre O with 
radius O P equal to 4 uoits describe the main lap circle 
L R P. The intersections of these two latter circles at Pand 
R give, as before explained (Case XVI.), the opening and 
closure respectively of the pore by the operation of the main 
valve, and it will be observed that upon the radius O 3 the 
width of the shaded crescent of port opening is 5 units, the 
prescribed amount. 

Upon O a:; as a diameter describe the expansion circle v x, 
and from the general centre O with 6 units radius, describe 
the slot circle N v M. The lens-shaped figure O v M. w, 
bounded by arcs of these two latter circles and shaded 
vertically, shows the slot opening, just as the crescent- 
shaped figure shaded horizontally shows the port opening. 

Upon the radius O 3 the intersection of the expansion circle 
occurs at the point t, 5 units from the centre O, showing the 
slot to be open that amount, and the slot is quite closed upon 
the radius O 4^, as that radius forms a tangent to the 
expansion circle, there being no intersection. 

It is easy to measure the amount of slot opening upon any 
given radms intermediate between O J where closing com- 
mences (shown by the intersection at v of the expansion and 
slot circles), and O 4^, by measuring from the centre O to the 
point where the given radius cuts the arc O v of the expansion 
circle. 

But port opening has to be measured upon the crescent- 
shaped figure from the lap circle, radially, and not from the 
centre 0'''; and to show the combiaed action of the valve aad 
the expansion plate in oae tigare, it is necessary to transfer 
or project the arc O v to 0' J, by drawing in a few temporary 
radii between O J and O 4^, and with a pair of dividers set to 
4 units (the radius of the lap circle) transferring the curve by 
t iking points upon each radius. The actual steam way during 
the wlule periad of admission (from O E, where the main 
valve opens the port, up to O 4^, where the expansioa plate 
•closes the slob) is thus represented by the figure in Diagram 
XXIII , euclosed by the black curves. 



* Because the main valve has lap ; but the expansion plate in the present 
'Case has none. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 45 

It will now be seen plainly that up to the radius O 3 the 
main valve controls the steamway, but from thence onwards 
the expansion plate takes charge, up to the point of cut-off 
upon the radius O 4^. (The main valve does not close the 
port UDtil the crank reaches O D.) 

In the diagrams which follow, this projection or transfer 
ence of an arc to the exterior of the lap circle will be adopted 
whenever necessary without going through all the inter- 
mediate steps. 

The diagram can, of course, be completed as regards the 
exhaust portion of it, from Cases XVI., XVII., or XXI. 

Case XXIV. — Conditions as in Case XXII., but with double 
OR multiple slotted expansion plate. To construct the 
valves and eccentrics for a given cylinder face ; the lap 
OF main valve, the lead, the point of cut-off, the width 
OF slots, and a position of the crank where a certain 
total slot opening is desired, being given. 

In order to reduce the travel of the expansion plate (which 
in case XXII. required a movement of nearly 42 units to 
accompl sh the given conditions), it is usual to employ double 
or multiple-slotted valves and platts, thus securing the same 
total slot opening with a reduced movement. 






^ 3 




y rf 4 


^ 


:j 



SCALE OF UNITS. 
Diagram 24. 



46 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



In diagram 24, the throw and angle of the main eccentric, 
the point of cut-off, and the desired steamway at a given 
position of the crank, correspond with those given for Case 
XXTI., but instead of a single slot of 6 units in width, the 
valve has two slots of 3 units each, and the expansion plate 
Jias a slot cut through it of equal width, so as to give two cut- 
off edges. 

Having, then, in the present case the centre of the main 
eccentric C, and its angle A O with the real position of the 
crank, fixed as in Case XXII., we proceed as in the latter 
part of that case, thus : — 

Draw the director Z C at right angles to the crank radius 
at which cut-off is to take place (in this case represented by 
O 4|) and produce it to X or further. From C as a centre, 
and with a radius equal to Imlf-^ the desired total slot opening 
a the given position of the c^ank (here 5 units total opening 
when the crank is on O 3, hence the radius will be ^^ units), 
descr be a short arc Y Y', and, touching this arc, draw the line 
W X at right angles to the given crank position (here O 3) 
produced. 

W X will cut Z X at X, which will be the centre of the 
expansion eccentric. Join X, f which will be the throw, 
and A O X will be the angle with respect to the real position 
of the crank at A of the expansion eccentric to fulfil the given 
conditions. 

Here C X is of course exactly half the length of C X in Case 
XXII., and were it not for the obstruction of the second or 




•.'>— 7? 18 



■i8-^5'4-^W 



■Scalt of Unys 

Valve with Double Slots and Series A Expansion Plate. 



FIG. 2, 



inner slot, due to the passing over it of the bar A of the ex- 
pansion plate (Fig. 2) we should, with our two 3 unit slots, 
get precisely the same total slot opening as in Case XXII. 
with the single 6 unit slot. 



* For a two-slotted valve. 
t The line O X is not shown on the diagrams. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 47 

We have now to find the extent and location of this 
obstruction. 

With centre O, and radius equal to C X, describe the ex- 
pansion travel circle y 7i x ; and with centre Z (the point of 
intersection of the cut-off radius O A¥) and radius equal to the 
width of slot (3 units), describe a small arc V V ; and from 
the same centre Z, with radius equal to ttvo slot widths (6 
units) describe a larger arc S U. 

Radii O H and O J drawn touching these arcs will represent 
respectively the positions of the crank when the second or 
inner slot begins to open, and when it attains its full opening 
of 3 units.''' 

The first or outer slot in the valve is of course subject to no 
such obstruction , and remains open during the entire semi-circle 
terminating at O 4^, the cut-off radius. Both slots begin to 
close when the crank is upon the radius O J ; both are open 
to the prescribed extent (2^ units) when the crank reaches 
O 3 ; and both of course close when the crank arrives at 
4i 

There is one more point to be considered before we can 
finish the construction of the valve and plate — viz., the length 
A (Fig. 2) of the bar, or interval of solid plate between the end 
of the expansion plate, and the slot (or, in a multiple-slotted 
plate, the interval between any two slots). 

This bar length A must in all cases be sufficient to ensure 
that the slot in the plate will not begin to uncover the outer 
slot in the valve before the port in the cylinder face has closed, 
or a reopening would occur during the period of expansion. 

This would be denoted by a hump in the lower part of the 
expansion curve in the indicator diagram. 

If the expansion plate have a variable travel, the bar length 
A must always be taken with reference to the longest travel 
of the plate relative to the main valve. 

Having found and laid down on the diagram the crank 
position O D, at which the valve closes the port in the cylinder 
face, the bar length may be easily found thus : — 

From the point Z, where the cut-off radius O 4,\ intersects 
the expansion travel circle ij ^L x, describe a short arc N N' 
touching the radius O D. 

The radius of this arc, plus the width of slot (in this case 
1\ units), is the length of bar A which would cause the reopen- 
ing to occur at the same moment that the main valve closed 
the port. 

* If the diagram has been correctly laid down, a small arc from centre 
Z, with radius of 2J units, should touch the crank radius O 3, checking the 
accuracy of the construction. 



48 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



Supposing that we allow a margin of safety of ^in. — i.e., 
that the port shall be over-closed to that extent before the 
slot is reopened,— the radius of the arc just mentioned would 
have to be increased by ^in. and the bar A would be made 
equal to this new radius plus the slot width. The radius O F, 
drawn to touch this new arc L L', would then represent the 
position of the crank when the slot harmlessly reopens. 



Chapter III —THE DOUBLE SLIDE VALVE. 

Diagram 24a (which is a duplicate of Diagram 24, save that 
the width of slots in valve and plate is increased by half-a- 
unit, and to which the same reference letters apply through- 
out) shows the effect of mcreasing the available steam way by 
widening the slots, retaining the same throw and angle of 




SCALE OF UNITS '^ 
Diagram 24a. 



expansion eccentric. It will be seen that the position of the 
crank O H, at which the second slot begins to open, occurs 
very much earlier in the revolution, while at O J, where both 
slots attain their full opening, the total slot width is 7 units 
an unnecessarily wide opening, as it is considerably in excess 
of the port opening at the same point. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



49 



b n c 



^^ 




'IS 24 30 36 4?- 

SCALE OF UNITS 



•54- 



Valve with Double Slots and Series A Expansion Plate (Slots 
b c widened, giving Negative Lap. to edge b). 



A better plan is to increase the width of the slot in the 
expansion plate, as shown in Fig. 3. Here, by increasing the 
width of slot 6 c in the expansion plate, retaining the original 
3-unit width for the slots in the valve, we can lessen the 
obstruction as desired. 

In Diagram 24b, which again is a copy of Diagram 24, except 
that the crank position O J, where the full opening of the 
second or inner slot commences, occurs much earlier in the 
revolution, the necessary width of slot in the expansion plate 




^CALE OF UNITS. 

Diagram 24b. 



50 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

b c, Fig. 3, is indicated by the radius of the arc V V struck 
from the centre Z, and touching the radius O J, where the 
full slot opening is to commence.* 

This construction is perfectly general, whether the slot b c 
in the plate be wider than the slots in the valve or not ; and 
in Diagrams 24 and 24a, where arcs struck from the centre Z 
touch the radius O J, the radii of these arcs will be found to 
coincide with the width of expansion-plate slots (which in 
those two diagrams are equal in width to their respective 
valve slots). 

But to proceed. A smaller arc vv' from the same centre Z, 
of 3 units radius (the slot width) will be observed to touch the 
radius O J'. This crank position O J' marks the beginning of 
slot closure. But while the crank is moving through the arc 
J J', the inner slot (owing to the negative lap of edge b) in 
the expansion plate remains wide open. 

Indeed, with the proportions here given the inner slot is not 
entirely obstructed during any part of the semi -circle ter- 
minating at O 4^, as was the case in the two previous 
examples. 

To test this assertion we have only to describe the arc S U 
from centre Z, with radius one slot width (3 units) in excess 
of the radius of the arc V V. It falls outside of the centre O, 
consequently the radius O H cannot be drawn. (In the two 
previous cases, O H marked the crank position at which the 
second or inner slot began to open). This shows that the slot 
does not quite close, the minimum opening being indicated by 
the aaiount the radius of the arc S U exceeds that of the 
expansion travel circle y Z x {in the present case about balf- 
a-unit). 

In the next case we shall have an opportunity of seeing 
whether the simple constructional Diagrams 24, 24a, and 24b, 
can be relied upon to fufil the given respective conditions. 

Case XXV.— Conditions as in Case XXIV. To construct 

THE ZEUNERf DIAGRAM, AND FROM IT THE VALVE AND EXPANSION 
PLATE ; AND TO DELINEATE THE CURVES OF PORT OPENING 

(steamway) FOR Diagrams 24, 24a, and 24b. 

As explained in Case XXIII., in considering the movement 
of the expansion plate by itself, we may neglect the motion 
of the main valve, and conduct our investigation as though it 
were a stationary face for the expansion plate to work upon. 

* If the valve had more than tvro slots, all the expansion-plate slots would 
be of this width b c. 

t Prof. Gustav Zeuner is not to be considered responsible for the extended 
applications of his system here introduced. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



51 



'' In the present case we have the centres of the main and 
expansion eccentrics fixed by Case XXIV., on the assumption 
that the lap of main valve is four units, the lead one unit, the 
point of cut-off is at ^ths of the stroke, and that when the 
crank is at fths of the stroke the steam way shall be five units. 
All these conditions are uniform with those assumed for Cases 
XXII. and XXIII ; but on the supposition that with the 
single slot of 6 units width, the construction in those cases 
gave too long a travel to the expansion plate, Case XXIV. 
shows the reduced movement applicable to a double-slotted 
valve to fulfil the given conditions. 




Diagram 25. 



52 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

In Diagram 52, we assume the width of each slot to be 
three units, and we shall now be able to see what effect this 
alteration will have upon the steamway. 

With Diagram 24 before us, we proceed to construct 
Diagram 25 as follows: — 

O C being the throw, and A O C the angle with respect to 
the real position of the crank at A, of the main eccentric ; 
describe upon O C as a diameter the main steam circle 
O R C P; and from centre O, with radius O P, equal to the 
prescribed lap of main valve (four units), describe an arc of 
the main lap circle R Q P. 

The intersections of these two circles at P and R give, as 
before explained (Case XVI.), the points of opening and 
closure respectively of the port, by the operation of the main 
valve, and the radii O E and O D, drawn cutting those points 
of intersection, give respectively the crank positions when 
the main valve opens, and again closes the port in the 
cylinder face. The crescent shaped figure R C P Q is not 
shaded in the present case, to avoid complicating the diagram ; 
but, as before, the breadth of the figure at any radius (say 
O 3, where the port opening was to be five units) represents 
the port opening when the crank occupies that radius. We 
may now dismiss the main valve for the present. 

From the general centre O, with radius C X (Diagram 24), 
describe the expansion travel circle y 7a x, and at right angles 
to the cut-off radius O 4^ draw the diameter y x oi the travel 
circle. Upon O x and O ?/ as diameters describe the expan- 
sion steam circles O Q a; and y L' O. From centre x, with 
radius x O, describe an arc U O S ; and from centre O, with 
radius equal to the slot width, describe the first slot circle 
N N' V M. Finally, from the same centre O, with radius 
equal to two slot widths, describe an arc of the second slot 
circle S T U, thus enclosing a lens-shaped figure O S T U. 

This lenticular figure is coincident in every way with the 
similar figure in case XXIII. \ O v ^ w^ save that a large 
part of its area left unshaded in the diagram, and forming a 
zone of the expansion circle O Q a;, represents the obstruction 
of the second or inner slot J by the passing over it of the bar 
A, Fig. 2, of the expansion plate. 

A moment's consideration of the double slotted valve and 
plate shown in Fig. 2, case XXIV., will make it clear that 
while the outer slot in the valve, when once uncovered by the 

t When the slot in the plate is wider than the valve-slots, this lenticular 
figure cannot be constructed. See Diagrams 24b and 25b. 

J The inner slot of the valve, although represented by the outer slot upon 
the diagram. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 53 

movement of the expansion plate, remains clear and unob- 
structed until the return of the plate closes it again at the 
appointed period of the stroke, the second or inner slot, by 
the passing over of the bar A, must at a certain point be 
obstructed or closed entirely, and again reopened some time 
before the actual cut-off point is reached. 

It becomes necessary, therefore, to exercise care in propor- 
tioning the valve and plate, so as to either keep this obstruc- 
tion within due limits, or remove it to a part of the stroke 
where its presence is unimportant. 

Referring now to the shaded parts of the lenticular figure 
representing slot opening, we see that upon the crank-radius 
O H the second slot begins to open, and is opened to its full 
width when the crank arrives at O J, at which point in the 
stroke both slots begin to close ; their combined width upon 
the radius O 3 measuring 5 units, the prescribed amount. 

All this is precisely in accordance with the conditions from 
which Diagram 24 was constructed, and when the breadths 
of shaded slot opening are collected and transferred upon 
their radii to the outside of the 4-unit main lap circle (as fully 
explained in Case XXIII.), we are able to see the actual 
steamway due to the combined action of valve and plate, 
through the whole period of admission, from O E, where the 
main valve opens (lead), up to O 4^, where the steam is cut 
off by the action of the expansion plate. This is represented 
by the area enclosed by the black lines. 

We have now to see whether the bar length A, Fig. 2, of 
the expansion plate, as found by construction in Diagram 24, 
is confirmed by the independent evidence of Diagram 25. 
We found that a bar length, equal to the slot width plus the 
radius of an arc struck from centre Z, touching the main cut- 
off radius O D, would cause the outer slot to be reopened 
simultaneously with the closing of the port by the main 
valve ; and we see by Diagram 25 that the arc N N' of the 
same radius § struck from centre O, cuts the intersection of 
the expansion circle y L' O by the radius O D. 

A second arc L L', of ^in. greater radius, designed to give 
a margin of safety of that amount, is reproduced as the arc 
L L' in Diagram 25, and shows the reopening (denoted by the 
shading) safely postponed until the crank arrives at O F. 

Perpendiculars dropped from L, from M, and from T in 
Diagram 25, construct the expansion plate full size without 

F § It happens in the present instance (and in Diagram 25b also) to coincide 
with the radius of the first slot circle, but this is merely a coincidence. In 
Diagram 25a this arc is shown independently as n n'. 



54 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

further trouble, and from it the slots 6, c, bar a 6 or A, and 
distance a c of the main valve, follow as a matter of course. 

Whether the steamway thus indicated is sufacient for the 
purpose of maintaining the full pressure upon the piston up 
to the point of cut-off is a question outside our present pur- 
pose ; the increase in the steamway may or may not keep 
pace with the increasing velocity of the piston, but we will 
assume that in the present instance it is insufficient, and that 
a wider or more rapidly-increasing steamway is desirable. ., 




SCALE or US/TS 
Diagram 25a. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 55 

One obvious method is to make the slots wider, retaining 
the same radius O x, and angle A O x oi the expansion 
eccentric, and Diagram 25a, to which the same letters of 
reference and description apply, shows the effect of an addi- 
tion of half a unit to the width of each slot in valve and 
plate. It will be at once seen that the crank arc during 
which the second slot is entirely closed, is very much 
shortened. At O J, where the two slots are for the moment 
wide open, their combined opening amounts to 7 units, which 
is a needlessly large amount, because, as the diagram shows, 
the port opening due to the main valve is only a little over 5 
units, when the crank occupies the same radius O J, 

This crank position O J, the point at which the obstruction 
of the second slot ceases, is important enough to merit a little 
consideration on its own account. 

In a valve and plate such as we have bepn describing, where 
the slots in both are of equal width, the position O J indicates 
the maximum of slot opening, because at the same moment 
the obstruction ceases closure commences. This maximum 
slot opening, in order to get the best effect from the com- 
bination, should obviously be placed with some regard to the 
point of maximum attainable port opening as determined by 
the action of the main valve. 

The maximum port opening always takes place when the 
crank is upon O C ; and when the cut-off point is later than 
this in the revolution, as in the present instance, we shall 
evidently be securing the best possible steam way under the 
given conditions by making O J coincide with O C. 

This, it may be remarked in passing, has nothing to do with 
the closing of the slots at a given position of the crank (in 
this case they were to be open 5 units at O 3, and closed at 
O 4^), which is a function of the throw and angle of the 
expansion eccentric O x. 

Supposing, therefore, that in Diagram 25 O J occurs too 
late in the stroke, and that we wish, while retaining the 3-unit 
slots in the valve, to get a wide-open second slot when the 
crank is at the point of maximum port opening O C. 

Diagram 25b is an exact copy of Diagram 25, with the single 
exception that the second slot is represented as radially 
distant from the first one a distance equal to the negative lap 
6 n of the slot in the expansion plate. The inner and outer 
arcs e M'^d and S T U' bounding this slot in Diagram 25b are 
struck from the centre O, with radii equal respectively to 
those of arcs V V and S U in Diagram 24b. The curves e S 
and d U' forming the ends of the figure are derived respectively 



56 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




SC^tLE OF HJUTS 



Diagram 25b. 

from the arcs O V and O 6 by projection or transference, 
employing the method explained in Case XXIII. 

Thus, on account of the interval between the two slot figures, 
they cannot be included within one lenticular figure, as was 
the case in Diagram 25 and 25a, but must be separately 
represented, the first by the small lens O V M 6, and the 
second by the arc-shaped figure bounded by the letters 
eSV d. 

Perpendiculars dropped from L, from M, and from T, in 
Diagram 25b, construct the expansion plate as before, the 
distance M M' in the diagram (= 6 n in the expansion plate) 
being the negative lap of the bach (or following) edge h of the 
slot h c (This must not be confounded with negative or minus 
lap of the cut-off edge of an expansion plate, of which examples 
will follow in due course). (Series C.) 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 57 

It will now be seen that, with the construction indicated by 
Diagram 24b, the slot opening fulfils exactly, as demonstrated 
in the present diagram, all the required conditions. The 
space within the black lines, showing, as before, the steamway 
due to the combined action of valve and plate, here shows the 
slot opening to be as nearly as possible equal to the port open- 
ing. This has been effected by the negative lap 6 n of the 
edge h of the plate allowing the slot which it controls to remain 
wide open while the crank passes from O J to O J', and the 
amount of such negative lap is always denoted in a diagram 
like 25b by the radial interval V f , M M', h n, separating the 
two slots. 

The length of bar a n in the valve is, as will be seen, equal 
to the length of bar in the plate, plus the negative lap, or 
a h + b n, which again is equal to the distance L M' on the 
base line of the diagram. 

The distance a c in the valve (= L T in the diagram) must 
of course coincide with ac in the plate, when both valve and 
plate are in their central position, as shown in Figs. 1, 2, and 
8, Series A. 

The diagram can, of course, be completed, as regards the 
exhaust portion of it, from Cases XVI., XVII., or XXI. 

Case XXVI. — Variable expansion by altering the angle 
OF the eccentric, its throw being fixed. Particulars of 
main valve and its eccentric being given, together with 
the maximum travel permissible for the expansion plate ;* 
to find the respective angles of the expansion eccentric 
for cutting off at given positions of the crank or of 
the piston. 

In the diagram let O C represent the throw, and A O C the 
angle with the crank, of the given main eccentric, and let O 5, 
4, 3, 2, and O 1 be the specified points of cut-off 
reckoned from B, the assuyned starting point of the crank (the 
real direction of rotation being indicated by the arrow). It is 
required to find the angles for the adjustable expansion 
eccentric, its throw being limited to a radius of the arc 

X^ X^, the expansion plate being of the Series A type, and 

the valve single slotted. 

At right angles to O 5 draw the director Z^C, and produce 
it to be cut the given arc. Its point of intersection X^ is the 
position of the centre of the expansion eccentric, and A O X^ 
its angle with the crank, for cutting off the steam when the 

* I.e., as limited by the inside length of the steam-chest, or by other 
conditions. 



58 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 
4 3 




crank occupies the radius O 5. In like manner the directors 

Z^C Z^C produced, cut the given arc at X* X^, and 

these denote the positions, and A O X^ A O X^ the angles 

with the crank, of the adjustable expansion eccentric for 
cutting off the steam when the crank occupies the respective 
radii O* OK 

It is thus an extremely simple matter to construct the 
diagram for variable expansion, the centre of the expansion 
eccentric being adjustable between the limits of X^ as its 
earliest position, and X^ as its latest. 

In all cases where the expansion plate is constructed to 
cut off with its outer edges, as in the present case, the ex- 
pansion eccentric leads — i.e., it is in advance of the main 
eccentric O C. 

Case XXVII. — Variable expansion by altering the angle 

OF THE ECCENTRIC, ITS THROW BEING FIXED. CONDITIONS AS 

IN Case XXVI. To construct the Zeuner diagram, and 

TO DELINEATE THE CURVES OF PORT OPENING (STEAMWAY) FOR 
THE SEVERAL POINTS OF CUT-OFF. 

In Diagram 27, the throw and angle A O C of the main 
eccentric, and the throw and angular positions of the ex- 
pansion eccentric for the given points of cut-off, are those 
given, or determined, by Case XXVI. 

The lap of the main valve is 4 units, and the lead is 1 unit ; 
the maximum permissible throw of the expansion eccentric is 
15 units ; the given points of cut-off are fths, i, fths, ^th, and 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



59 



^th of the piston's stroke (reckoned from the assumed starting 
point B), and the valve has a single slot of 6 units width. It 
is required to construct a diagram showing the exact amount 
of steamway open (for each position of the expansion eccentric) 
at any crank position from O up to the point of cut-off. 




Diagram 27. 

With centre and radius O C describe the main travel 
circle A D C E, and upon O C as a diameter, describe the 
main steam circle O B, C P. Again, from centre O, with radius 
O P equal to 4 units, describe the main lap circle L R P. Tha 
intersections of these two latter circles at P and R mark the 
opening and closing of the port by the action of the main 
valve (Case XVI.) ; and the radii O E and O D, passing 
through P and R, denote the position of the crank at each 
operation respectively. 

Now join C X^ C X^ C X^ (Case XXVL), and draw 

O x^ O x^ equal and parallel each to each. Upon O x^..^ 

...O x^ as diameters, describe their respective expansion 
circles.* 

* In the diagram, semi-circles only for the sake of clearness. 



'60 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

Each of these circles will touch the corresponding crank 
radius representing the point of cut-off for that circle. Thus, 
taking the fths cut-off as an example, we see that its (semi-) 
•circle O t v w x^ is tangential to the radius O^ showing the 
slot opening to be zero ; at O* the slot is open a distance O t ; 
at O^ the slot is open a distance O v ; at O^ it is open a distance 
O w ; and finally at x^ the slot is open to its full extent of 
■6 units, as shown by the diameter O x^ reaching the circle of 
slot -opening N M (which is 6 units radius). 

Each of the other semi-circles shows its corresponding slot- 
opening in the same way, and it will be noticed that the 
largest semi-circle (that for the |^th point of cut-off) — though 
this is a coincidence merely — intersects the slot circle at 
exactly the same point — viz., upon the radius O a;^ - as the 
semi-circle for the fths cut-off does, showing that in the two 
extreme positions of the expansion eccentric, slot closing 
begins at the same position of the crank. 

The dark curves exterior to the main lap circle are simply 
the projections or transferences of the corresponding arcs of 
the expansion circles within the lap circle, which thus com- 
bined with the shaded crescent representing ^or^opening 
show at a glance the actual steamway at any position of the 
crank due to the united action of the main valve and the ex- 
pansion plate. Taking the outer or phs curve as an example, 
it will be seen how simply this is effected by setting a pair of 
dividers to 4 units (the lap of the main valve), transferring 
the points O to O^, t to f-, v tov''-,w to w^, etc., and drawing in 
a curve passing through the points so found. 

In this way the action of different valves and valve gears 
•can be readily compared, and the most suitable for the 
requirements selected, with an accuracy which leaves nothing 
to be desired, and with the least possible expenditure of time 
and trouble. 

In the present case we have a single-slot valve, and no 
question of reopening can arise (see remarks at the end of 
•Case XXV.). But in the case of variable expansion with a 
•double-slot valve, care must be taken that the slots are so far 
apart in the valve that at no position of the eccentric can a 
reopening occur. 

Assuming for the moment that there were two slots in the 
valve in the present case, their minimum distance apart would 
be found thus : — 

In Diagram 27 make O y'^ equal and opposite to O x^, 
iihe largest expansion circle. Upon O ^ as a diameter 
•describe a circle (or a portion of a circle, as the dotted arc at y^). 
Produce the radius of main travel circle O D, which is the 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



61 



position of the crank when the main valve clones, to a, where 
it cuts the dotted arc. By making the distance between the 
slots (see A, Fig. 2, Case XXIV.) equal to O a, Diagram 27, 
plus the width of slot,* the expansion plate will begin to re- 
open the outer slot in the main valve at the same moment 
that the latter closes the port in the cylinder face. By making 
the distance between slots equal to O & + slot width, we en- 
sure a margin of safety to the extent otab (which may be ^in. 
or whatever is considered necessary) against reopening. 

The diagram should be completed as regards the exhaust 
portion of it from Cases XYI., XVII. or XXI., Part I. 

Case XXVIII. — Variable expansion by altering the throw 
OF the eccentric, its angle being fixed. Particulars of 

MAIN VALVE AND ITS ECCENTRIC BEING GIVEN, TOGETHER WITH 
THE MAXIMUM TRAVEL PERMISSIBLE FOR THE EXPANSION PLATE ; 
TO FIND THE RESPECTIVE THROWS OF THE EXPANSION ECCENTRIC 
FOR CUTTING OFF AT GIVEN POSITIONS OF THE CRANK OR OF 
THE PISTON. 

In Diagram 28 let O C represent the throw, and A O C tho- 
angle with the crank, of the given main eccentric, and let 
O 5, O 4, O 3, O 2, O 1, and O Ot be the specified points of 
cut-off, reckoned from B, the assumed starting point of the- 




-12 



SCALE OF L'MTS 
Diagram 28. 



* I.e., the lap == O a 

\I.e., cut-oflE at zero for automatic expansion gear. 



62 THE PROPOETIONS AND MOVEMENT OF SLIDE VALVES. 

crauk (the real direction of rotation being indicated by the 
arrow). It is required to find the respective throws for the 
variable expansion eccentric, its maximum throw being in the 
present case supposed limited to 15 units, and the expansion 
plate being of the Series A type. 

With centre O, and radius equal to 15 units, the specified 
maximum throw, describe a short arc a b, and at right angles 
to O 5, the latest desired point of cut-off, draw the director 
Z^ C, and produce it to cut the short arc a b. Its point of 
intersection X^ is the position of the centre of the expansion 
eccentric, and A O X^ its angle with the crank for cutting off 
the steam when the crank ocjupies the radius O 5. 

Join O X^ and draw the directors Z^ G Z° C, and produce 

them to cut the radius O X^. The points of intersection with 

X^— viz., X*, X^ X'^ — are the successive centres of the 

variable expansion eccentric, for cutting off at the respective 
points specified, the eccentric being movable at the constant 
angle A O X^ between the Umits of X° as its earliest position, 
and X^ as its latest. 

Case XXIX. — Variable expansion by altering the throw 

OF THE eccentric, ITS ANGLE BEING FIXED. CONDITIONS AS IN 

Case XXVIII. To construct the Zbuner diagram, and to 
delineate the curves of port-opening (steamway) for the 
several points of cut-off. 

In Diagram 29, the throw and angle A O C of the main 
eccentric, and the angular position and throws of the expan- 
sion eccentric for the given points of cut-off, are those given 
or determined by Case XXVIII. 

The lap of the main valve is, as before, 4 units, the lead is 

1 unit, and the port -width is, of course, 6 units. 

The maximum permissible throw of the expansion eccentric 
is 15 units. The given points of cut-off are f ths, ^, f ths, ^th, 
l^th, and O of the piston's stroke (reckoned from the assumed 
starting point B), and the valve has a single slot of 6 units 
width. It is required to construct a diagram showing the 
exact amount of steamway open (for each separate throw of 
the expansion eccentric) at any crank position from O up to 
the point of cut-off. 

With centre O and radius O C describe the main travel 
circle A D C E, and upon O C as a diameter describe the 
main steam circle O R C P. Again from centre O, with radius 
O P, equal to 4 units, describe the main lap circle L R P. 

The intersections of these two latter circles at P and R 
mark the opening and closing of the port by the action of the 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 63 




SCALE OF UNITS 

Diagram 29. 

main valve (Case XVI.) ; and the radii O E and O D, passing 
through P and R, denote the position of the crank at each 
operation respectively. 

Now, join C X^ C X* C X", and draw O x^ O x'^ equal 

and parallel each to each. Upon O x^ O x^ as diameters, 

describe their respective expansion (semi-) circles. 

Each of these circles will touch, tangentially, the corres- 
ponding crank radius representing the point of cut-off for that 
circle. Thus, taking the fths cut-off as an example, we see 
that its (semi-) circle O t v w x^ \^ tangential to the radius 
O 5, showing the slot opening to be zero. At O 4 the slot is 
open a distance O ^ ; at 3 the slot is open a distance O v ; 
at O 2 it is open a distance O w ; and, finally, at O x^ the slot 
is open to its full extent of 6 units, as shown by the diameter 
x^ reaching the circle of slot-opening N M (which is 6 units 
radius). 

Each of the other semi-circles shows its corresponding 
slot-opening in the same way, and it will be noticed that the 
semi-circle described upon the diameter O x'^, and touching 
tangentially the radius O B (showing the slot to be closed at 
the beginning of the piston's stroke), is yet open a littlet 
when the main valve begins to unclose the port upon the 
radius O E, thus giving pre-admission, though cutting off at 



t Shown by the filled-up angle at P. 



64 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

a 

The dark curves exterior to the lap circle are, as before, 
simply the projections or transferences of the corresponding 
arcs of the expansion circles within the lap circle, which, 
thus combined with the shaded crescent representing port- 
opening, show at a glance the actual steamway at any position 
of the crank, due to the united action of the main valve and 
the expansion plate. Taking the outer or fths curve as an 
example, it will be seen how simply this is effected by setting 
a pair of dividers to 4 units (the lap of the main valve) and 
transferring the points O to O', t to t', v to v', w to w', etc., 
and drawing in a curve passing through the points so found. 

By comparing the curves in the present Diagram 29 with 
those of Diagram 27, the respective steamways obtained by 
the two alternative methods of producing variable expansion 
can be readily contrasted, as the proportions are 'identical in 
the two cases. 

The diagram should be completed, as regards the exhaust 
portion of it, from Cases XVI., XVII., or XXI., Part I. . _ ..- 



CHAPTEB IV. — Expansion plate to cut off with its inner 
edges, and having no lap, either positive or negative. 

The form of expansion plate which we propose to examine 
next differs from the previous type (Series A) in being 
arranged to close the slots by means of its inner edges, a, a, 
Fig. 1. 

As a consequence of this modification the eccentric by 
which it is driven has to be arranged to follow the main 
eccentric instead of leading, as in the previous examples. 

Case XXX. — To construct the main and expansion valves 

AND ECCENTRICS FOR A GIVEN CYLINDER FACE ; THE LAP OF MAIN 
VALVE, THE LEAD, THE POINT OF CUT-OFF, THE WIDTH OF THE 
SLOT, AND A POSITION OF THE CRANK WHERE A CERTAIN SLOT 
OPENING IS DESIRED BEING GIVEN. 

Upon the base-line A B, with centre O and radius O P equal 
to the amount of lap, describe the main lap-circle L S P, and 
lay off upon the line A B, P Q e lual to the given lead. tFpon 
A B at the point Q erect the lead-line Q C as a perpendicular ; 
and let O 3 represent the position of the crank (supposed to 
start from B) where the port and the slot are both to be open 
the given amount. 

Lay off, upon the crank radius O 3, the distance S T, 
exterior to the lap circle, equal to the given port-opening. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



65 



From the point T draw the line T C at right angles to O T, 
cutting the perpendicular Q C at C. Join O C, which will be 
the throw of the main eccentric; and the angle A O C will be 
its angle with respect to the real position of the crank at A 
(Case VII.). 




Diagram 30. 



Let O 4y represent the position of the crank where the 
expansion plate is to cut off the steam. Draw O y at right 
angles to O 4|^ ; and upon O 3 produced, lay off a distance 
O Z^ equal to the given sZo^opening when the crank is at O 3. 
Draw Z^ ?/ at right angles to O 3 produced, cutting Oyaty. 
O y would be the throw, and A. O y the angle with respect to 
the real position of the crank at A, of an eccentric which 
would fulfil the given conditions as to cutoff and slot-opening 
if the main slide valve were fixed, forming a stationary face 
for the expansion plate to work upon. 

But in the present (as in the previous) series, the expansion 
plate is required to effect the prescribed conditions as to cut- 
off and slot-opening upon a moving main valve driven by an 
eccentric, whose throw and angular position with respect to 
the crank are denoted by A O C. 

All that is necessary to accomplish this movement is to 
lefer the line O ^ to the point C as an origin. Thus C X is to 
be made parallel with and equal to O y. 



66 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

The point X is the centre of the eccentric for moving the 
expansion plate upon the back of the main valve (whose 
eccentric centre is at C) to fulfil the given conditions ; and, 
joining O X, we have A O X as its angular position with 
regard to the real position of the crank at A. 

This can be done in an even simpler manner by construc- 
tion direct from the centre of the main eccentric C, thus (see 
dotted lines on Diagram 30) : — 

Having fixed the point C so as to fulfil the prescribed con- 
ditions, as in the earlier part of the present cass, or otherwise, 
draw the director Z C '^' at right angles to O 4^, and produce 
it to X or further. From C as a centre, and with radius equal 
to the desired slot opening when the crank is upon O 3, 
describe a short arc W Y, and, touching this arc, draw the 
line Y X at right angles to O 3 ; Y X will cut Z X at X, which 
will be the centre of the expansion eccentric as before. t 




Valve with Single Slots and Series B Expansion Plates. Fig. 1. 

In designing a Series B valve and plate, the distance apart 
of the slots in the upper surface of the valve (B, Fig. 1), and 
the length A of the bars composing the expansion plate itself, 
must receive careful consideration. If the distance B be 
insufficient, the right-hand slot may be partially covered by 
the left-hand bar when the plate reaches the extremity of its 
travel to the right ; or vice-versa. 

The minimum distance B necessary to secure an unob- 
structed slot is one-half the actual movement of the plate 
upon the back of the valve or C X, Diagram 30. 

* Director : See Case XXII., last paragraph. 

t In Diagram 30 the dotted line Y X touches the main lap-circle at S. This, 
is merely an accidental coincidence. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 67 

Again, if the bar-length A, Fig. 1, be insufficient, the slot 
may be reopened by the passing over of the bar, before the 
main valve closes the port, giving a readmission of steani 
during the period of expansion. The minimum length of bar 
A necessary to avoid this may be found from Diagram 30^ 
thus : — 

From the general centre O, with radius O ?/ (or C X), describe 
the expansion travel circle y z x. Let O D represent the 
position of the crank when the main valve closes the port. 
From the point z, where the cut-off radius 4^ intersects the 
expansion travel circle y z x, describe a short arc N N^ 
touching the radius O D. The radius of this arc, plus the 
ividih of slot (in this case six units), is the length of bar A, 
which would cause the reopening to occur simultaneously with 
the closure of the port by the main valve. A margin of safety 
is desirable, and for this purpose ^in., whether the valve be a 
large or a small one, is not too much. If, now, another arc, 
n n^, of, say, ^in., greater radius, be struck from centre z, the 
crank radius O F, drawn to touch it, will indicate the later 
position of the crank when reopening occurs, with a bar-length 
A equal to the slot-width plus the radius of arc n v?-. 

Case XXXI. — Conditions as in Case XXX. To construct 

THE ZeUNER diagram, AND FROM IT THE VALVE AND EXPANSION 
PLATE, AND TO DELINEATE THE CURVE OF PORT OPENING 
(STEAMWAY). 

In Diagram 31 the throw and angle A O C of the main 
eccentric, and the throw and angle A O ^Z of the supposititious 
eccentric for moving the expansion plate upon the back of a 
fixed main valve, are those found by Case 30. The lap of the 
main valve is 4 units ; the lead is 1 unit ; the point of cut-off 
is ^ths of the piston's stroke, reckoned from the assumed 
starting point B (or, as the stroke is divided into eighths, at 4|- 
eighths of the stroke). The width of port is 6 units, and the 
width of the slot in the main valve is 6 units also ; and finally 
the steamway, or the clear opening through both slot and 
port, is to be 5 units at f ths of the stroke. 

These conditions are precisely those laid down for the 
corresponding Case 23, Series A. 

Having Diagram 30 before us, we proceed to construct 
Diagram 31 thus :— 

With centre O and radius O C describe the main travel 
circle A D C E, and upon O C as a diameter describe the main 
steam circle O R C P ; and again from centre O, with radius 
P equal to 4 units, describe the main lap circle L R P. 



68 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




sv4i.£ ffr ttmrs 

Diagram 31. 



The intersections of these two latter circles at P and R give 
as before explained (Case 16), theopeniog and closure respec- 
tively of the port by the operation of the main valve. If the 
diagram has been laid down correptly, the width of the 
crescent enclosed by the arcs R C P and II O^ P will be 5 units 
upon the radius O 3, which is the prescribed port opening for 
that position of the crank. 

Produce y O to x, and make O x equal and opposite to O y. 
Upon :r as a diameter describe the expansion circle O v x, 
and from the general centre O, with 6 units radius, describe 
the slot circle N v M. The lens-shaped figure O v M w, 
bounded by arcs of these two latter circles and shaded 
vertically, shows the slot opening, just as the crescent -shaped 
figure shaded horizontally shows the port opening. 

Upon the radius O 3 the intersection of the expansion circle 
occurs at the point t, 5 units from the centre O, showing the 
slot to be open that amount ; and the slot is quite closed upon 
the radius O 4^, as that radius forms a tangent to the expan- 
sion circle, there being no intersection. 

It is easy to measure the amount of slot-opening upon any 
given radius intermediate between O J, where closing 
commences (shown by the intersection at v of the expansien 
and slot circles), and O A.^, by measuring from the centre O 
to the point where the given radius cuts the arc O v of the 
expansion circle. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 69 

The simple method by which the arc O v is projected or 
transferred to the exterior of the main lap circle is fully ox- 
plained in Case 23, and need not be repeated here. The 
figure enclosed by the black lines represents the actual steam- 
way during the whole period of admission from O E, where 
the main valve opens the port, up to O 4^, where the 
expansion plate closes the slot ; and the width of the figure 
upon any radius within those limits is the measure of the 
steamway actually open at the moment the crank occupies 
that radius. 

We have now to see whether the bar length A (Fig. 1) of 
the expansion plate, as found by construction in Diagram 30, 
is confirmed by the independent evidence of Diagram 31. 

We found that a bar length equal to the slot width plus the 
radius of an arc N N^ struck from centre z, touchmg the main 
cut-off radius O D, would cause the slot to be reopened simul- 
taneously with the closing of the port by the main valve ; and 
we see by Diagram 31 that its arc N^ N of the same radius, -'' 
struck from centre O, cuts the intersection of the expansion 
circle y n^ O with the radius O D. It will thus be seen that 
two separate operations are happening at the instant the 
crank occupies the radius O D. 

The main valve (having lap equal to O R) is closing the 
port, evidenced by the intersection at R of its lap circle and 
steam circle, and the bar of the expansiou plate (assumed for 
the moment to have foUoiuing la^pf equal to O N^), having 
passed over the slot, is reopening it, as deuoted by the inter- 
section at N^ of the arc N N^ with the expansion circle, the 
intersections at R and N^ both occurring upon the line O D 

To postpone this reopening until the crank has reached the 
position O F, we make the following lap of bar A equal to 
O n} {n^ being the point where O F cuts the expansion circle). 

Perpendiculars dropped from n, O, and M, in Diagram 31, 
construct the bar of the expansion plate full size without 
further trouble, O n^ being the lap and O M the slot width, 
together making up the length A. (See also Fig. 1). Th3 
minimum distance B between the bars of the expansion plate, 
given in Diagram 30 as equal to C X, (=0 y in Diagram 31), 
is always half the maximum actual movement of the expan- 
sion plate upon the back of the main valve. 

The diagram can, of course, be completed, as regards the 
exhaust portion of it, from Cases 16, 17, or 21. 

* Arc N^ X here happens to coincide with the slot-circle N v M. This is 
merely accidental and has no significance. 

t Following lap— i.e., the lap of its following or trailing edge. See O n% 
Fig. 1. The leading edge A has no lap. 



70 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



Case XXXII. — Conditions as in Case XXX., but with 

DOUBLE OR MULTIPLE-SLOTTED EXPANSION PLATE. To CONSTRUCT 
THE VALVES AND ECCENTRICS FOR A GIVEN CYLINDER FACE, THE 
LAP OF MAIN VALVE, THE LEAD, THE POINT OF CUT-OFF, THE 
WIDTH OF THE SLOTS, AND A POSITION OF THE CRANK WHERE A 
CERTAIN TOTAL SLOT-OPENING IS DESIRED, BEING GIVEN. 

The double- slot ted main valve with its expaasion plate for 
the Series B type is shown in Fig. 2. 



K- A'-^ UA-i.--B 




■SCALE Of lIMjfS- 
Valve with Double Slots and Series B Expansion Plate. Fig. 2. 



Series B, it will be remembered, is an expansion plate con- 
structed to cut off with its inner edges, and having no lap, 
either positive or negative. As explained in Case XXX., the 
expansion eccentric has to be set to follow the main eccentric 
instead of leading it, as in the Series A cases. 

In Diagram 32 the throw and angle of the main eccentric, 
the point of cut-off, and the desired steamway at a given 
position of the crank, correspond with those given for Case 
XXX. ; but instead of a single slot of 6 units in width, the 
valve has two slots of 3 units each, and the expansion plate 
has a slot cut through it of equal width, so as to give two cut- 
off edges. 

Having then in the present case the centre of the main 
eccentric C and its angle A O C with the real position of the 
crank fixed as in Case XXX., we proceed as in the latter part 
of that Case, thus : — 

Draw the director Z C at right angles to the crank radius 
at which cut-off is to take place (in this case represented by 
O 4^), and produce it to X, or farther. From C as a centre 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



71 




SCALP &r UHlTS 
DIAGRAM 32. 

and with radius equal to half^ the desired total slot-opening 
(here 5 units total opening when the crank is on O 3, hence 
the radius will be 2|- units) describe a short arc Y Y\ and 
touching this arc draw the line Y X at right angles to O 3. 
Y X will cut Z X at X, which will be the centre of the 
expansion eccentric. Join O X, which will be the throw, and 
A O X will be the angle with respect to the real position of 
the crank at A of the expansion eccentric to fulfil the given 
conditions. 

Here C X is, of course, exactly half the length of C X in 
Case XXX. ; and were it not for the obstruction of the second 
or outer slot, due to the passing over it of the bar A of 
the expansion plate Fig. 2, we should, with our two 3 unit 
slots, get precisely the same total slot-opening as in Case 
XXX. with the single 6-unit slot. 

We have now to find the extent and location of this 
obstruction. 

With centre O and radius equal to C X describe the expan- 
sion travel circle y z x \ and with centre z (the point of inter- 
section of the cut-off radius O 4^) and radius equal to the 
width of slot (3 units) describe a small arc V V^, and from the 
same centre z with radius equal to two slot-widths (6 units) 
describe a larger arc S U. 

Radii O H and O J, drawn touching these arcs, will repre- 
sent respectively the positions of the crank when the second 



For a two-slotted valve. 



72 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

or outer slot begins to open and when it attains its full 
opening of three units.* 

The first, or inner, slot in the valve, provided the space B 
(Fig. 2) between the bars A be not less than O v/, Diagram 32 
(or one-half the movement of the expansion plate upon the 
main valve), is of course subject to no such obstruction, and 
remains open during the entire semi-circle terminating at 
O 4^, the cut-off radius. Both slots begin to close when the 
crank is upon the radius O J ; both are open to the prescribed 
extent (2^ units) when the crank reaches O 3, and both of 
course close when the crank arrives at O 4^. 

There is one more point to be considered before we can 
finish the construction of the valve and plate — viz., the length 
of the bars A and A\ Fig. 2.t 

This bar length must in all cases be sufficient to ensure 
that the slots in the valve shall not be uncovered or reopened 
by the passing over of the bars until the port in the cylinder 
face has been safely closed by the main valve. This would 
allow a readmission of steam during the period of expansion, 
and would be denoted by a hamp in the lower part of the 
expansion curve in the indicator diagram. 

If the expansion plate have a variable travel, the bar length 
A must always be taken with reference to the longest travel 
of the plate, relative to the main valve. 

Having found and laid down on the diagram the crank 
position O D at which the main valve closes the port in the 
cylinder face, the bar length may be easily found thus : — 

From the point z, where the cut-off radius O 4| intersects 
the expansion travel circle y z x, describe a short arc N N^ 
touching the radius O D. 

The radius of this arc plus the width of slot (in this case 2^ 
units), is the length of bar A, which would cause the reopen- 
ing to occur at the same moment the main valve closed the 
port. 

Supposing that we allow a margin of safety of ^in.—^.e., 
that the port should be over-closed to that extent before the 
slots reopen — the radius of the arc just mentioned would have 
to be increased by ^in., and the length of the bars would be 
made equal to this new radius plus the slot width. The 
radius O F drawn to touch this new arc L L^ would then 
represent the position of the crank when the slots harmlessly 
r eopen. 

* If the diagram has been correctly laid down, a small arc from centre z, 
with radius of 2J units, should touch the crank radius O 3, checking the 
accuracy of the diagram. 

_t These bars are precisely similar ; the length we shall arrive at for bar A 
will apply also to bar A\ 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 7o 

Diagram 32a is a duplicate of Diagram 32, save that the 
width of slots in valve and plate is incrf ased by half a unit. 
The same reference letters apply throughout. The arc V V^, 
struck from centre Z with radius equal to the slot width of 
3^ units (instead of 3 units), shows by its tangent the crank 
radius O J, the earlier position of the crank at which both 
slots attain their full opening. The arc S U, with radius of 
two slot widths (7 units), here falls beyond the centre O, so 
that the crank radius O H (marking the beginning of opening 
for the second slot) cannot be drawn as its tangent. 




scale of units 
Diagram 32a. 



This shows that the second or outer slot does not quite 
close during any part of the semi-circle terminating at O 4^ 
{i.e., is not} completely obstructed), the minimum opening 
being indicated by the amount the radius of the arc S U 
exceeds that of the expansion circle y 7a x. 

The full total slot opening at O J is here 7 units, which is 
unnecessary, as it is in excess of the port opening at the 
same point. 

A better plan is to increase the width of the slot in the 
expansion plate, as shown in Fig. 3. 

Here, by increasing the width of slot 6 c in the expansion 
plate, retaining the original width of 3 units for the slots in 
the valve, we can lessen the obstruction as desired. 



74 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




1 24 "3 0-: — 36—42- 

SCALE, or VNtTS. 



.48 — .54 ^^^60 



Valve with Double Slots and Serifs B Expansiotv Plate (Slot b e 
widened, giving negative lap to edge b). 

Fig. 3. 

In Diagram 32b (which agaia is a copy of Diagram 32, 
except that the crank position O J, where the fall opening of 
the second or outer slot commences, occurs much earlier in 
the revolution) the necessary width of slot in the expansion 
plate h c, Fig. 3, is indicated by the radius of the arc V V^ 
struck from centre Z, and touching the radius O J, where the 
full slot-opening is to commence.-'' 

This construction is perfectly general, whether the slot h c 
in the plate bo wider than the slots in the valve or not ; and 
in Diagrams 32 and 32a, where arcs struck from the centre Z 




-.ALL c^ u^srs 
Diagram 32b. 



* If the valve had more than two slots, all the slots in the plate would be of 
this width, b c. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 75 

touch the radius O J, the radii of these arcs will be found to 
coincide with the width of expansion plate slots (which in 
these two cases are equal to their respective valve slots). 

A smaller arc v v^ from the same centre Z of 3 units radius 
(the slot width), will be observed to touch the radius O J^. 
This crank position O J' marks the beginning of slot-closure. 
But while the crank is moving through the arc J JS the outer 
slot (owing to the negative lap of edge h in the expansion 
plate) remains wide open. 

As was the case in the preceding Diagram 32a, the outer 
slot is not entirely obstructed during any part of the semi- 
circle terminating at O 4^.t 

To test this assertion we have only to describe the arc S U 
from centre Z, with radius one slot width (3 units) in excass of 
the arc V V^. It falls outside the centre O, consequently the 
radius O H cannot be drawn. (In Case XXXII., O H marks 
the crank position at which the second or outer slot began to 
open.) This shows that the slot does not quite close, the 
minimum opening being indicated by the amount the radius 
of arc S U exceeds that of the expansion circle y "L x (in the 
present case about half a unit). 

In the next case we shall have an opportunity of seeing 
whether the simple constructional Diagrams 32, 32a, and 32b 
can be relied upon to fulfil the given respective conditions. 

Case XXXIII. — Conditions as in Case XXXII. To con- 
struct THE ZeUNErJ diagram ; AND FROM IT THE VALVE AND 
expansion PLATE, AND TO DELINEATE THE CURVE OF PORT- 
OPENING (STEAMWAY) FOR DIAGRAMS 32, §2a, AND 32b. 

As explained in Case XXIII. ,§ in considering the movement 
of the expansion plate by itself, we may neglect the motion 
of the main valve, and conduct our investigation as though it 
were a stationary face for the expansion plate to work upon. 

In the present case we have the centres of the main and 
expansion eccentrics fixed by Case XXXII. On the assump- 
tion that the lap of main valve is 4 units, the lead 1 unit, the 
point of cut-off is at ^ths cf the stroke (or 4^ eighths), and 
that when the crank is at f ths of the stroke the steamway 
shall be 5 units. 

All these conditions are uniform with those assumed for 
Cases XXX. and XXXI. ; but on the supposition that with the 
single slot of 6 units width the construction in those cases 

t See Diagram 33b. 

t See the first footnote, Case XXV. 

§ See the second footnote, Case XXIII, (substituting A O 2/ for A O x.) 



76 



THE PROPORTIONS AND MOVEMENT OP SLIDE VALVES. 



gave too long a travel to the expansion plate, Case XXXII. 
shows the reduced movement applicable to a double-slotted 
valve to fulfil the given conditions. 

In diagram 25 we assume the width of each slot to be 3 
units, and we shall now be able to see what effect this altera- 
tion will have upon the steamway. 

With Diagram 32 before us, we proceed to construct Dia- 
gram 33 as follows : — 

O C being the throw and A O G the angle, with respect to 
the real position of the crank at A, of the main eccentric, 
describe upon O C as a diameter the main steam circle, 
O R C P, and from centre O, with radius O P equal to the 
prescribed lap of the main valve (4 units), describe an arc of 
the main lap circle R Q P. 




.SCAJil or UNITS 

Diagram 33. 



The intersections of these two circles at P and at R give, as 
before explained (Case XVI.), the points of opeoing and 
closure respectively of the port by the operation of the main 
valve ; and the radii O E and D, drawn cutting those points 
of intersection, give respectively the crank positions when 
the main valve opens, and again closes, the port in the 
cylinder face. The crescent-shaped figure R C P Q is not 
shaded in the present case, to avoid complicating the diagram ; 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 77 

but, as before, the breadth of the figure at any radius (say, 
O 3, where the port-opening was to be 5 units) represents the 
port-opening occupying that radius. We may now dismiss 
the main valve for the present. 

From the general centre O, with radius C X (Diagram 32), 
describe the expansion travel circle y Ti x; and at right angles 
to the cut-off radius O 4^ draw the diameter y x ot the travel 
circle. Upon O x and O ?/ as diameters describe the expan- 
sion steam circles O Q :i; and y L^ O. From centre x, with 
radius x O, describe an arc U O S ; and from centre O, with 
radius equal to the slot width, describe the first slot circle 
N N^ V M. Finally, from the same centre O, with radius 
equal to two slot widths, describe an arc of the second slot 
circle S T U, thus enclosing a lens-shaped figure O S T U. 

This lenticular figure is coincident in every way with the 
similar figure in Case XXXI., O v M. w,\\ save that a large 
part of its area (left unshaded in the diagram, and forming 
a zone of the expansion circle O Q x) represents the obstruc- 
tion of the second or outer slot^ by the passiog over it of the 
bar A, Fig. 2, of the expansion plate. 

A moment's consideration of the double-slotted valve and 
plate shown in Fig. 2 (Case XXXII.) will make it clear that 
while the inner slot of the valve when once uncovered by the 
movement of the expansion plate remains clear and un- 
obstructed (provided the space B Fig. 2 and Diagrams 33, 33a, 
and 33b be not less than O y, or half the movement of the 
plate upon the valve) until the return of the plate closes it 
again at the appointed period of the stroke, the second, or 
outer, slot, by the passing over it of the bar A, must at a 
certain point be obstructed, or closed entirely, and again re- 
opened some time before the actual cut-off point is reached. 

It becomes necessary, therefore, to exercise care in pro- 
portioning the valve and plate so as to either keep this 
obstruction within due limits, or remove it to a part of the 
stroke where its presence is unimportant. 

Referring now to the shaded parts of the lenticular figure 
representing slot-opening, we see that upon the crank radius 
O H the outer slot begins to open, and is open to its full width 
when the crank arrives at O J, at which point in the stroke 
both slots begin to close ; their combined width upon the 
radius O 3 measuring 5 units, the prescribed amount. 

I! When the slot in the plate is wider than the slots in the valve, this len- 
ticular figure cannot be constructed. See Diagrams 32b and 33b. 

^ In this case the outer slot of the valve is represented by the outer slot on 
the diagram. 



78 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

All this is precisely in accordance with the conditions from 
which Diagram 32 was constructed ; and when the breadths 
of shaded slot -opening are collected and transferred upon 
their radii to the outside of the 4-unit main lap-circle (as 
fully explained in Case XXIII.), we are able to see the actual 
steamway due to the combined action of valve and plate, 
through the whole period of admission, from O E, where the 
main valve opens (lead), up to O 4^, where the steam is cut 
off by the action of the expansion plate. This is represented 
by the area enclosed by the black lines. 

We have now to see whether the bar length A of the 
expansion plate, as found by construction in Diagram 32, is 
confirmed by the independent evidence of Diagram 33. We 
found that a bar length equal to the slot width plus the radius 
of an arc struck from centre Z, touching the main cut-off 
radius O D, would cause the outer slot to be reopened simul- 
taneously with the closing of the port by the main valve, and 
we see by Diagram 33 that the arc N^ N of the same radius* 
struck from centre O cuts the intersection of the expansion 
circle ?/ L' O by the radius O D. 

A second arc L L^ of ^in. greater radius, designed to give a 
margin of safety of that amount, is reproduced as the arc 
L L^ in Diagram 33, and shows the reopenicg (denoted by the 
shading) safely postponed until the crank arrives at O F. 

Perpendiculars dropped from L, from M, and from T, in 
Diagram 33, construct the expansion plate full size without 
further trouble ; and from it the slots b c, bar a b, or A, and 
the distances a c, and B (or O y), follow as a matter of course. 

Whether the steamway thus indicated is sufficient for the 
purpose of maintaining the full pressure upon the piston up 
to the point of cut-off, is a question outside our present 
purpose ; the increase in the steamway may or may not keep 
pace with the increasing velocity of the piston ; but we will 
assume that in the present instance it is insufficient, and that 
a wider or more rapidly-increasing steamway is desirable. 

One obvious method is to make the slots wider, retaining 
the same radius O y, and angle A O y, ot the expansion 
eccentrio ; and Diagram 33a, to which the same letters of 
reference and description apply, shows the effect of an addition 
of half a unit to the width of each slot in valve and plate. 

It will be at once seen that the outer slot is not entirely 
obstructed, as was the case in Diagram 33. The shading 

* It happens in the present instance, and in one or two other cases, to coin- 
cide with the radius of the first slot-circle N V M ; but this is merely a 
coincidence. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



19 



shows the opening to be continuous during the semi-circle 
terminating at O 4^. Although the minimum opening at x is 
very small, at O J, where the two slots are for the moment 
wide open, their combined opening amounts to 7 units, which 
is a needlessly large amount, because, as the diagram shows, 
the port-opening due to the main valve is only a little over 5 
units when the crank occupies this position O J. 

This crank position O J, the point at which the obstruction 
of the second or outer slot ceases, is important enough to 
merit a little consideration on its own account. 





i^-^^mmm^: m^/(m. 



I //y / / / / yx.^/yy" y/ A 



SCyli4 OP VUITS. 

Diagram 33a. 



In a valve and plate such as we have been describing, where 
the slots in both are of equal width, the position O J indicates 
the maximum of slot-opening, because, at the same moment 
obstruction ceases closure begins. This maximum slot-opening, 
in order to get the best effect from the combination, should 
obviously be placed with some regard to the point of maximum 
attainable ^or^opening, as determined by the action of the 
main valve. 

The maximum port-opening always takes place when the 
crank is upon O C, a,nd when the cut-off point is later than 
this in the revolution, as in the present instance, we shall 
evidently be securing the best possible steamway, under the 
given conditions, by making O J coincide with O C. 



80 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



This, it may be remarked in passing, has nothing to do 
with the closing of th-e slot at a given position of the 
crank (in this ci.se it was to be 5 units at O 3), which is a 
function of the throw and angle of the expansion eccentric 
Oy. ^ 

Supposing, therefore, that in Diagram 33 J occurs too late 
in the stroke, and that we wish, while retaining the 3 unit 
slots in the valve, to get a wide-open second or outer slot 
when the crank is at the point of maximum port-opening O C. 

Diagram 83b is an exact copy of Diagram 33, with the single 
excepnion that the second oc outer slot is represented as 
radially distant from the firs'; one a distance equal to the 
negative lap b n ot the slot in the expansion plate. The inner 
and outer arcs e M^ d and S T U^ bounding thi«( slot in 
Diagram 33b, are struck from the centre O, witti radii equal 
respectively to those of arcs V V and S U in Diagram 32b. 
The curves e S and d U\ forming the ends of the figure, are 
derived respectively from the arcs O V and O b, by projection 
or transference, employing the method explained in Case 
XXIII. 




SOALt OF UOITS 

Diagram 33b. 



"^ Thus, on account of the interval between the two slot 
figures, they cannot be included within one lenticular figure, 
as was the case in Diagrams 33 and 33a, but must be 



THE PEOPORTIONS AND MOVEMENT OF SLIDE VALVES. 81 

separately represented ; the first by the small lens O V M &, 
and the second by the arc-shaped figure bounded by the 
letters e S U^ fZ. 

Perpendiculars dropped from L, from M, and from T in 
Diagram 33b, construct the expansion plate as before ; 
the distance M M^ in the diagram (= h n in the expansion 
plate) being the negative lap of the bacJc {or following) edge b 
of the slot b c. This must not be confounded with negative 
or minus lap of the cut-off edge of an expansion plate, of 
which examples will be given in the next series (C). 

It will now be seen that with the construction indicated by 
Diagram 32b, the slot-opening fulfils exactly, as demonstrated 
in the present diagram, all the required conditions. The 
space within the black lines, showing as before the steamway 
due to the combined action of valve and plate, here shows the 
slot- opening to be as nearly as possible equal to the port- 
opening. This has been effected by the negative lap b n of 
the edge b of the plate, allowing the slot which it controls to 
remain wide open while the crank passes from O J to O J^ ; 
and the amount of such negative lap is always denoted, in a 
diagram like 33b, by the radial interval V y, M M, b n, 
separating the two slots. 

The length of bar a n in the valve is, as will be seen, equal 
to the length of bar in the plate plus the negative lap, or 
a b + b n-, which, again, is equal to the distance L M^ on the 
base line of the diagram. 

The distance a c in the valve ( =: L T in the diagram) must 
of course coincide with a c in the plate when both valve and 
plate are in their central position, as shown in Figs. 1, 2, and 
3, Series B. 

The diagram can of course be completed, as regards the 
exhaust portion of it, from Cases XVI., XVII., or XXI. 

Case XXXIV. — Variable expansion by altering the angle 
OF the eccentric, its throw being fixed. Particulars of 

MAIN valve and ITS ECCENTRIC BEING GIVEN, TOGETHER WITH 
THE MAXIMUM TRAVEL PERMISSIBLE FOR THE EXPANSION PLATE ;'- 
TO FIND THE RESPECTIVE ANGLES OF THE EXPANSION ECCENTRIC 
FOR CUTTING OFF AT GIVEN POSITIONS OF THE CRANK OR OF THE 
PISTON ; AND TO CONSTRUCT THE EXPANSION PLATE. 

In the diagram let O C represent the throw, and A O C the 
angle with the crank of the given main eccentric ; and let 
O 5, O 4, O 3, 02, and O 1 be the specified points of cut-off, 

* I.e., as limited by the inside length of the steamchest, or by other 
conditions. 



82 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



reckoned from B, the assumed starting point of the crank 
(the real direction of rotation being indicated by the arrow). 
It is required to find the angles for the adjustable expansion 
eccentric, its throw being limited to a radius of the arc X^ 

X^ ; the expansion plate being of the Series B type, and 

the Talve single-slotted. 

At right angles to O 5 draw the director Z^ C and produce 
it to cut the given arc. Its point of intersection X^ is the 
centre of the expansion eccentric, and A O X^ its angle with 
the crank for cutting off the steam when the crank occupies 

the radius O^. In like manner the directors Z^ C Z' C, 

produced, cut the given arc at X^ X^ ; and these denote 

the positions, and A O X^ A O X^ the angles with the 

crank, of the adjustable eccentric for cutting off the steam 
when the crank occupies the respective radii O* O^ 

It is thus an extremely simple matter to construct the 
diagram for variable expansion, the centre of the expansion 
eccentric being adjustable between the limits of X^ as its 
earliest position and X^ as its latest. 

In all cases where the expansion plate is constructed to cut- 
off with its inner edges, as in the present case, the expansion 
eccentric /o^^ows the main eccentric O C. 




SCALE Of fMTS. 

Diagram 34, 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 83 

In designing a Series B valve and plate for variable expan- 
sion, eare must be taken that neither obst7'icction nor reopening 
of the slot can take place in any position to which the expan- 
sion eccentric may be adjusted. 

Referring to Fig. 1, Case XXX., the minimum distance B 
between the inner edges a, a of the bars forming the expan- 
sion plate will be equal to C X^, Diagram 34, this being one 
half the actual maximum movement of the plate upon the 
back of the valve. If the distance B be less than this, the 
right-hand slot may be partially covered by the left-hand bar 
when the plate reaches the extremity of its maximum travel 
to the right, or vice-versa. 

Again, if the bar length A (Fig. 1) be insufficient, the slot 
may be reopened by the passing over of the bar before the 
main valve has closed the port. This would cause a readmis- 
■sion of steam during the period of expansion. The minimum 
length of bar A necessary to avoid this may be found from 
Diagram 34, thus : — 

From the general centre 0,with radius C X^ (the maximum 
throw or half travel of the plate upon the back of the valve), 
describe a short arc y 7i. From the point Z, the intersection 
of the maximum cut-off radius O 5 produced, describe a short 
arc N N\ touching the radius O D produced (O D being, as in 
all previous cases, the crank position where the main valve 
closes its port). The radius of this arc N N^, plus the widtk 
of slot (in this case 6 units), is the length of bar A, which at 
the plate's maximum travel would cause the reopening of the 
slot to occur simultaneously with the closure of the port by 
the main valve. 

If we wish to postpone this reopening until the crank arrives 
at the position O F, the radius of arc L^ L drawn touching 
O F produced, will be the amount which, added to the width 
of slot,t will give the correct length of bar A for the maximum 
and all lesser travels. 

Case XXXV. — Variable expansion by altering the angle 

OF THE ECCENTRIC ; ITS THROW BEING FIXED. CONDITIONS AS 

IN Case XXXIV. To construct the Zeunbr Diagram, and 

FROM it the valve AND EXPANSION PLATE ; AND TO DELINEATE 
THE CURVES OF PORT-OPENING (STEAMWAY) FOR THE SEVERAL 
POINTS OF CUT-OFF. 

In Diagram 35, the throw and angle A O C of the main 
eccentric, and the throw and angular positions of the expan- 
sion eccentric for the given points of cut-off, are those given 
in or determined by Case XXXIV. 

t I.e., the radius of arc L' L is the lap of the ba^k or following edge of the 
har( = On, Fig. 1). 



84 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 85 

The lap of the main valve is 4 units, and the lead is 1 unit ; 
the maximum permissible throw of the expansion eccentric 
is 13 units ; the given points of cut-off are fths, ^, fths, ^th, 
and ^th of the piston's stroke (reckoned from the assumed 
starting point B), and the valve has a single slot of 6 units 
width. It is required to construct a diagram showing 
the exact amount of steamway open (for each position of the 
expansion eccentric) at any crank position from O up to the 
point of cut-off. 

With centre O and radius O C describe the main travel 
circle A D C E, and upon O C as a diameter describe the main 
steam circle O R C P. Again from centre O, with radius O P 
equal to 4 units, describe the main lap circle L R P. 

The intersections of these two latter circles at P and R 
mark the opening and closing of the port by the action of the 
main valve (Case XVI.) ; and the radii O E and O D passing 
through P and R denote the position of the crank at each 
operation respectively. 

Now join C X^ C X4 C X^ (Case XXXIV.) and make O C^ 

equal and opposite to O C (the throw of the main eccentric). 

With centre O C^, and radius equal to the given throw 
of the expansion eccentric (here 13 units) describe the arc 

2/^, ij^ 2/^. Radii O y^, O i/ O i/, equal and parallel to 

C X^, C X* C X^, and terminating in this arc, are the 

throws, and A O y^, A O y^ A O y^, the angles with the 

crank, of the supposititious eccentrics for moving the expansion 
plate upon the back of a fixedj main valve. 

With centre O C (the throw of the main eccentric) and 
radius equal to the given throw (13 units) of the expansion 

eccentric, describe the arc x^,x^ x\ and, terminating in 

this arc, draw the radii O x^, O x^ O x^, equal and parallel 

to C X^, etc. (These radii will be, of course, equal and 
opposite to O y^, etc.) Upon O x^, etc., as diameters, describe 
their respective expansion circles. § 

Each of these circles will touch tangentially the corres- 
ponding crank radius representing the point of cut-off for 
that circle. Thus, taking the five-eighths cut-off as an example, 
we see that its (semi-) circle (the largest) is tangential to the 
radius O 5, showing the slot-opening to be zero. At O 4 the 
slot is open a distance O ^ ; at Of the slot is fully open 
(denoted by the intersection of the slot circle N i) M with the 
expansion circle). 

Each of the other semi-circles shows its corresponding slot- 
opening in the same way. It may be observed in passing that 

I See footnote Case XXIII., page 43, substituting A O y for A O ic. 
§ In the diagram, semi-circles only, for the sake of clearness. 



86 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

this Series B expansion plate is well adapted for cases like 
the present, where the angle of the expansion eccentric is 
adjustable for variable expansion, the throw being constant. 
This is evident from the circumstance that the expansion 
circles increase in size as the point of cut-off becomes later. 
(Compare the Series A expansion plate in the corresponding 
Case XXVII., where the expansion circles decrease the later 
the cut-off occurs.) 

On the other hand, this Series B arrangement is not well 
suited for variable expansion by altering the throw of the 
expansion eccentric with the angle constant, the slot-openings 
being insufficient and the travel of the expansion plate for a 
zero cut-off excessive. Consequently the cases for this form 
of variable expansion do not occur in this series. 

The dark curves exterior to the main lap circle are simply 
the projections or transferences of the co»rrespondlng arcs of 
the expansion circles within the lap cirdle ; which, thus com- 
bined with the shaded crescent representing port-opening, 
show at a glance the actual steamway at any position of the 
crank, due to the united action of the main valve and the 
expansion plate. The exact method of doing this has been 
described in several previous cases, ^ and need not be repeated 
here. 

By comparing these dark-lined curves of steamway with 
the similar lines of Case XXYII., the fuller openings given by 
the present arrangement (Series B) are shown at a glance. 

We have now only to see whether the bar length A (Fig. 1) 
of the expansion plate, as found by construction in Diagram 
34, is confirmed by the independent evidence of Diagram 35. 

We found that a bar length equal to the radius of an arc 
N Ni, Diagram 84, plus the slot width (6 units), would cause 
the slot to be reopened simultaneously with the closing of the 
port by the main valve ; and we see, by Diagram 85, that its 
arc N Ni of the same radius struck from centre O exactly 
crosses the intersection of O D with the expansion circle 
described upon O y^, the maximum throw (or one-half the 
maximum travel) of the expansion plate upon the main 
valve, f This shows that the two operations really do occur 
at the moment the crank is upon O D. The main valve 
(having lap equal to O B) ig' closing the port, evidenced by 
the intersection at R of its lap circle and steam circle, and 

* See Case XXIII. and others. 

+ It must be noted that while the real throw O X of the expansion eccentric 
in this case is constant, the virtual throw O y of the plate upon the valve 
varies. (It is the longest virtual throw O y^ which must be used in deter- 
mining; the proportions of valve and plate.) 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



87 



the bar of the expansion plate (assumed for the moment to 
have length equal to O N^ plus the slot width, 6 units) having 
passed over the slot, is reopening it, as denoted by the inter- 
section at Ni of the arc N N^ with the largest expansion circle, 
the intersections at R and N^both occurring upon the line O D. 

To postpone this reopening until the crank has reached the 
position O F, we make the bar length equal to O n^ plus the 
slot width (ni being the point where O F cuts the expansion 
circle). 

Perpendiculars dropped from n, from O, and from M in 
Diagram 35 construct the bar of the expansion plate full sfize, 
without further trouble, O n^ being the following lap-'', and 
O M the slot width, together making up the length A. (See 
also Fig. 1.) 

The minimum distance B between the bars of the expansion 
plate, given in Diagram 34 as equal to C X'^ (=03^^ in Diagram 
35), is always half the maximum virtual travel of the expan- 
sion plate. 

The diagram can, of course, be completed, as regards the 
exhaust portion of it, from Cases XVI., XVIL, or XXI. 



CHAPTER V. 

Expansion plate to cut off with its outer edges, 

AND having negative OR MINUS LAP. 

The valve and plate which form the subject of this chapter 
are, save in one respect, identical with the Series A type 
illustrated and described in Chapter III. The difference con- 
sists in the expansion plate being shorter than the outer edges 




SCALE OF UA'ITS 

Valve with Single Slots and Series C Expansion Plate. 
Fig. 1. 



See footnote Case XXXI., page 6y. 



b8 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

of the slot by a distance A (see Fig. I.) The letter A will be 
used to denote the amount of this minus lap'-' or space, which 
is commonly, though not necessarily, equal to the width of 
slot. 

Case XXXVI.— To construct the main and expansion 

VALVES AND ECCENTRICS FOR A GIVEN CYLINDER FACE. ThE 

LAP OF MAIN VALVE, THE LEAD, THE POINT OF CUT-OFF, THE 
WIDTH OF THE SLOT, AND A POSITION OF THE CRANK WHERE A 
CERTAIN SLOT-OPENING IS DESIRED, BEING GIVEN. 

Upon the base line A B with centre O, and radius O P equal 
to the amount of lap, describe the lap circle L S P, and lay off 
upon the line A B, P Q equal to the given lead f Upon A B, 
at the point Q, erect the lead-line Q C as a perpendicular, 
and let O 3 represent the position of the crank (supposed to 
start from B) where the port (as well as the slot) is to be open 
the given amount. 




Diagram 



Lay off, upon the line O 3, the distance S T, exterior to the 
lap circle, equal to the given port-opening. From the pomt 
T draw, perpendicular to O 3, the director J T C, cutting the 

* The term minus lap, though involving a technical contradiction, is con- 
venient and expressive. 

t Lead. See footnote to Case XXII. 

t Director, a perpendicular to a radius of the crank circle, drawn to C, 
X, or X 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 89 

lead-line Q C at C. Join O C, which will be the throw of the 
main eccentric ; and the angle A O C will be its angle with 
respect to the real position of the crank at A (Case VII.). 

Now let O 4^ represent the position of the crank where the 
steam is to be cut off by the action of the expansion plate ; 
and upon O 4^ produced lay off O A equal to the negative or 
minus lap (a, Fig. 1) of the expansion plate. Draw the 
director at a , and produce it to x or further. From the point 
A measure towards O the distance A a^, equal to the amount 
the slot is to be open at the given position O 3. Thus the 
distances a a^ and S T are by the conditions of this case 
equal. 

From centre O, with radius O a^, describe a circle or part 
of a circle, and touching this circle, draw perpendicular to O 3 
produced the director-tangent ai x. The point of intersection 
of these two directors, a x and Ai x will be the throw, and 
(joining O x) the angle h.0 x will be the angle with respect 
to the real position of the crank at A of an eccentric which 
would fulfil the given conditions as to cut-off and slot-opening, 
if the main slide valve were fixed, forming a stationary face 
for the expansion plate to work upon. 

But in the present (as in the previous) series, the expansion 
plate is required to effect the prescribed conditions as to cut- 
off and slot-opening upon a moving main valve, driven by an 
eccentric whose throw and angular position with respect to 
the crank are denoted by A C. 

All that is necessary to accomplish this movement is to 
refer the line O ;i; to the point C as an origin. Thus C X is to 
be made parallel with and equal to O x. 

The point X is the centre of the eccentric for moving the 
expansion plate upon the back of the main valve (whose 
eccentric centre is C) to fulfil the given conditions, and joining 
O X we have A O X as its angular position with regard to the 
real position of the crank at A. 

This can be done in an even simpler manner by construction 
direct from the centre of the main eccentric C, thus (see 
dotted lines in Diagram 36) : — 

Having fixed the point C so as to fulfil the prescribed con- 
ditions, as in the earlier part of the present case, or otherwise, 
proceed as follows : — With centre C and radius C a equal to 
the minus lap of the expansion plate, describe the arc of the 
expansion lap circle a Y. Touching this arc, and perpendic- 
ular to the given crank radius O 4|^, where steam is to be cut- 
off, draw the director at Z, and produce it to X or further. 
Upon the radius O C lay off a a ^ equal to the desired slot 
opening when the crank is at the given position O 3, and with 



90 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



centre C and radius C a^ describe an arc. Touching this arc, 
and perpendicular to the crank position O 3 (where the slot 
is to be open an amount a a^), dravv^ the director at W, and 
the point X where the two directors intersect is the centre of 
the expansion eccentric as before. 

Case XXXYII.— Conditions as in Case XXXVI. To con- 
struct THE Zeuner Diagram, and to delineate the curve 
OF port-opening (steamway). 

In Diagram 37, the throw and angle A O C of the main 
eccentric, and the throw and angle A O :r of the supposititious 
eccentric for moving the expansion plate upon the back of a 
fixed § main valve, are those found by Case XXXVI. The 
lap of the main valve is 4 units, the lead is 1 unit. The point 
of cut-off is ^ths of the piston's stroke, reckoned from the 
assumed startiog point B (or, as the stroke is divided into 
eighths, at ^^ eighths of the stroke). The width of port is 6 
units, and the width of the slot in the main valve is 6 units 




^^CALf. Oh UNjr^ 

Diagram 37- 



\ See second footnote to Case XXIII. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 91 

also; and, finally, the steam way, or the clear opening 
through both slot and port, is to be 5 units at fths of the 
stroke. These are the given conditions, and Diagram 37 will 
soon show whether the throws and angles found by Diagram 
36 can be relied upon to fulfil them accurately or not. 

First of all, with centre O and radius O C, describe the 
main travel circle A D G E, and upon O C as a diameter 
describe the main steam circle O K C P ; and. again, from 
centre O, with radius O P equal to 4 uuits, describe the main 
lap circle L R P. The intersectioos of these two latter circles 
give, as before explained (Case XVI.), the opening and closure 
respectively of the port by the operation of the main valve, 
and it will be observed that upon the radius O 3 the width of 
the (horizontally) shaded crescent of port opening is 5 units, 
the prescribed amount. 

Make now O y equal and opposite to O ;i; in Diagram 36, 
and upon O ^ as a diameter describe the expansion circle 
O IV y o^ (which, with this class of expansion plate, is upon 
the left of the diagram), and finally, from the general centre 
O, with radius of 6 units (equal to the slot width), describe 
the slot circle N v M w. The crescent shaped figure bounded 
by arcs of these two latter circles, and shaded vertically, 
shows the sZoi-opening, just as the before mentioned horizon- 
tally-shaded crescent shows the port-opening. 

Now in this case, in order to arrive at the curve of slot- 
opening, we measure distances, not from the centre O, as in 
the corresponding Case XXIII., (where the expansion plate 
had no lap, negative or otherwise), but radially inwards from 
the circumference of the slot circle N v M w. 

Thus, upon the given radius O 4^, where cut-off is to take 
place, these two circles intersect at o^, and there is evidently 
no opening— the slot is closed. Upon the other given radius 
O 3, where the prescribed slot-opening is 5 units, the breadth 
of the vertically-shaded crescent — viz., from V- to t — will be 
found to scale 5 units. As will be seen, the slot is open 
through nearly two-thirds of the revolution beginning at w, 
while through the half-circle extending from O Z^ to O Z, the 
slot is open its full width of 6 units (Z^ Z being a line drawn 
perpendicular to y x). 

Now we have very clearly, by means of the horizontally 
and vertically shaded crescents of Diagram 37, the respective 
openings of port and slot shown separately for any position of 
the crank within their range. In order to combine these two 
into one figure which shall show the actual clear steamway 
during the whole period of admission (from the crank position 
O E, where the main valve begins to open the port, to the 



92 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



crank position O 4^, where cut-off occurs by the agency of the 
expansion plate) all that is required is the transference or 
projection of the arc o^ O (by drawing in a few temporary 
radii between O Z and O 4^, and upon each radius transferring 
the radial breadth of the shaded portion) to the exterior of 
the main lap circle L K P. Thus upon radius O 3 the breadth 
f t is transferred to T 3, and so on, giving points in a curve 
I Z, which is thus derived from the arc o^ O. An examination 
of the j&gure enclosed by the thick black lines in Diagram 37 
will show that up to the crank position O 3 the main valve 
controls the steamway, but from thence onward the expansion 
plate takes charge, up to the point of cut-off. (The main 
valve does not close the port until the crank reaches the 
position O D). 

The diagram can, of course, be completed as regards the 
exhaust portion of it, from Cases XVI., XVII., or XXI. 

Case XXXVIII.— Conditions as in Case XXXVI., but with 

DOUBLE OR MULTIPLE-SLOTTED VALVE To CONSTRUCT THE 

VALVES AND ECCENTRICS FOR A GIVEN CYLINDER FACE. ThE 
LAP OF MAIN VALVE, THE LEAD, THE POINT OF CUT OFF, THE 
WIDTH OF THE SLOTS, AND A POSITION OF THE CRANK WHERE A 
CERTAIN TOTAL SLOT-OPENING IS DESIRED, BEING GIVEN. 

The double-slotted main valve, with its expansion plate for 
the Series C type, is shown in Fig. 2. 




1 2 18 24 30 36 4 2- 

SC>t.LE OF VJVITS 



Valve with Double Slots and Series C Expansion Plate. Fig. 2. 

In Diagram 38, the throw and angle of the main eccentric, 
the point of cut-off, and the desired steamway at a given 
position of the crank, correspond with those given for Case 
XXXVI. ; but instead of a single slot of 6 units in width, the 
valve has two slots of 3 units each, and the expansion plate 
has a slot cut through it of equal width, so as to give two cut- 
off edges. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 93 




SCALE OF UflJTS 

Diagram 38. 

Having then, in the present case, the centre of the main 
eccentric C, and its angle A O C with the real position of the 
crank fixed as in Case XXXVI., we proceed as in the latter 
part of that case, thus : — 

With centre C and radius C a equal to the minus lap of the 
expansion plate (a, Fig. 2), describe the arc of the expansion 
lap circle Y a Y^ Touching this arc, and perpendicular to the 
given crank radius O 4^, where steam is to be cut off, draw 
the director and produce it to X or further. Upon the radius 
O C lay off A A^ equal to lialf^ the desired total slot opening 
(here suppo&ed 5 units when the crank is on O 3, hence the 
distance a a^ will be 2| units), and with centre C and radius 
Ca^ describe an arc. Touching this arc, and perpendicular 
to the crank position O 3 (where each slot is to be open the 
prescribed amount a a^), draw the director and produce it to 
X, the point of intersection with the director previously drawn. 
This point X will be the centre of the expansion eccentric. 
Join O X,! which will be the throw, and A O X will be the 
angle with respect to the real position of the crank at A of 
the expansion eccentric to fulfil the given conditions. 

Here C X is of course exactly half the length of C X in 
Case XXXVI., and were it not for the obstruction of the 
second or inner slot, due to the passing over it of the bar A 
of the expansion plate Fig. 2, we should, with our two three- 
unit slots, get precisely the same total slot-opening as in Case 
XXXVI. with the single six-unit slot. 

* For a two-slotted valve, 
t The line O X is not drawn in on the diagram. 



94 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

We have now to find the extent and location of this 
obstruction. 

With centre O, and radius equal to C X, describe the expan- 
sion travel circle y z x, and with centre 4^ (the point of inter- 
section of the cut-off radius O 4^ with the expansion travel 
circle) and radius equal to the minus lap a of the expansion 
plate (3 units), describe an arc V V^. Touching this arc, draw 
the radius O z, which is the position of the crank (supposed 
to start from B) when both slots are wide openj — i.e., the 
expansion plate is exactly in mid travel with respect to the 
maia valve upon which it moves. 

Now with centre z and radius of 3 units describe the semi- 
circle U S H. The radius O 4^, which this semi-circle will 
touch, denotes, as we know, the position of the crank when 
both slots are just closed ; and similarly the radius O H, also 
touched by the semi-circle, marks the position of the crank 
when the second or inner slot begins to open. 

The first or outer slot is of course subject to no obstruction, 
and remains wide open during the half revolution terminating 
at z. 

Both slots begin to close when the crank is on the radius z ; 
both are open to the prescribed extent (2^ units) when the 
crank reaches O 3, and both of course close when the crank 
arrives at O 4|. 

There is one more point to be considered before we can 
finish the construction of the valve and plate — viz., the length 
A (Fig. 2) of the har, or interval of solid plate between the 
end of the expansion plate and the edge of the first slot (or in 
a multiple-slotted plate, the interval between any two slots). 

This bar length A must in all cases be sufficient to ensure 
that the slot in the plate will not begin to uncover the outer 
slot in the valve before the port in the cylinder face has 
closed, or a reopening would occur during the period of 
expansion. 

This wo aid be denoted by a hump in the lower part of the 
expansion curve in the indicator diagram. 

If the expansion plate has a variable travel the bar length 
A must always be taken with reference to the longest travel 
of the plate relative to the main valve. || 

Having found and laid down upon the diagram the crank 
position O D, at which the main valve closes the port in the 
cylinder face, the bar length may be easily found thus : — 

t If the diagram has been correctly laid down, a small arc from centre z, 
with radius of half a unit, should touch the crank radius O 3, where the slots 
are to be each open the prescribed amount of 2J units. 

II See latter part of Case XLI. and following casss. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 95 

From the point z in the expansion travel circle y z x (where 
the crank position O z, at the moment when the expansion 
plate is in mid-travel, intersects it) describe a short arc N N^ 
touching the radius O D. 

The radius of this arc is the length of the A which 
would cause the reopening to occur at the same moment that 
the main valve closed the port. 

Supposing that we allow a margin of safety of ^in. — ix.. 
that the port shall be overdosed to that extent before the 
slot is reopened, — the radius of the arc just mentioned would 
have to be increased by ^in. and the bar A would be made 
equal to this new radius. The radius O F, drawn to touch 
this new arc L L\ would then represent the position of the 
crank when the slot harmlessly reopens. 




.SCALE OF Vf*ITS. 

Diagram o8a. 

Diagram 38a (which is a duplicate of Diagram 38, save that 
the slot widths in valve and plate, and the minus lap a , are 
all increased by half a unit) is constructed precisely as in the 
previous diagram, and shows the effect of increasing the 
available steamway by widening the slots. As before, the 
cut-off occurs at the crank position O 4^, and the slots are 
open 2| units when the crank is upon O 3. The throw and 
angle of the expansion plate will be found to differ a little, 
but the reference letters of Diagram 38 apply throughout. 

The crank position O H, at which the second slot begins to 
open, occurs, as will be seen, somewhat earlier in the revolu- 
tion ; while at O z, where both slots are for the moment wide 



96 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



open, their combined opening is 7 units, an unnecessarily wide 
opening, as it is considerably in excess of the port-openinj?, 
at the same point. 



a A, Un c 



mf i>^^>^m>gs^ i^ 




W//m///m ^ ^2m ^ 




m^ 




-'^CALE or. vr^'irs 

Fig, 3— Valve with Double Slots and Series C Expansion Plate, 
(SLOT b c widened, giving negative following LAP to edge b). 

A better plan is to increase the width of the slot in the 
expansion plate as shown in Fig. 3. Here, by increasing the 
slot width by an amount h n, we can lessen the obstruction as 
desired, while retaining the 3-unit slots in the valve, and the 
throw and angle of expansion eccentric, of Diagram 38. 

In Diagram 88b, which again is a copy of Diagram 38, 
except that the crank position O H, where the second or inner 
slot begins to open, occurs much earlier in the revolution, the 
necessary width of slot & c in the expansion plate Fig. 3 is 




SCALE OF Uf^ITS 
Diagram 38b. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 97 

indicated by the radius of the arc S H, struck from the centre 
z, and touching the crank position H, where the openiog of 
the second slot is commenced.''' 

This construction is perfectly general, whether the slot h c 
in the plate be wider than the slots in the valve or not ; and 
in Diagrams 38 and 38a, where arcs S H struck from the 
centre z touch the radius O H, the radii of these arcs will be 
found to coiccide with the width of the expansion plate slots^ 
which in those two diagrams are equal in width to their res- 
pective valve slots. 

But to proceed. An arc S^ H^ from centre z, with radius 3^ 
units (or a , the minus lap of the expansion plate) less than, 
the radius of arc S H, will indicate the position O C of the crank 
when the second slot is fully open. It remains wide open 
until the crank reaches O z, when closure of both slots com- 
mences. So that if we wish the second slot to remain wide 
open while the crank is moving, say, from O C to O 5;, we have 
only to make & n on the expansion plate equal to radius of 
arc S^ H^, drawn to touch crank position O C on the diagram. 
The remainder of Diagram 38b, from O z onwards, differs in 
no respect from Diagram 38. 

In the next case we shall have an opportunity of seeing 
whether the simple constructional diagrams, 38, 38a, and 38b, 
can be relied upon to fulfil the respective given conditions. 

Case XXXIX.— Conditions as in Case XXXVIII. To 
Construct the ZEUNERt Diagram, and from it the valve 

AND expansion PLATE ; AND TO DELINEATE THE CUR\ES OF 
PORT-OPENING (sTEAMWAY) FOR DIAGRAMS 38, 38a, AND 38b. 

As explained in Case XXIII, in considering the movement 
of the expansion plate by itself, we may neglect the motion of 
the main valve and conduct our investigation as though it 
were a stationary face for the expansion plate to work upon. 

In the present case we have the centres of the main and 
expansion eccentrics fixed by Diagram 38, on the assumption 
that the lap of main valve is four units, the lead one unit, 
the point of cut-off is at 4|- eights (or ^) of the stroke, and 
that when the crank is at f of the stroke the steamway shall 
be five units. All these conditions are uniform with those 
assumed for Cases XXXVI, and XXXVII. ; but on the supposi- 
tion that with the single slot of six units' width the construc- 
tion in those cases gave too long a travel to the expansion 

* If the valve had more than tvFO slots, all the expansion plate slots would 
be of this width, h c. 
t See footnote to Case XXV 



98 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



plate, Case XXXVIII. shows the reduced movemeat applicable 
to a double-slotted valve to fultil the given conditions. 

In Diagram 38 we assume the width of each slot to be three 
units, and with that diagram before us we proceed to construct 
Diagram 39, as follows : — 




SCALE OF VNTTS 
Diagram 39. 



O C being the throw, and A C the angle with respect to 
the real position of the crank at A, of the main eccentric : 
describe upon O C as a diameter the main steam circle 
O R C P ; and from centre O, with radius O P, equal to the 
prescribed lap of main valve (four units), describe an arc of the 
main lap circle R Q P. 

The intersections of these two circles at P and R give, as 
before explained (Case XVI.), the points of opening and 
closure respectively of the port, by the operation of the main 
valve, and the radii O E and O D, drawn cutting those points 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 99 

of intersection, give respectively the crank positions when the 
main valve opens, and again closes the port in the cylinder 
face (the crank being assumed to start from B on the diagram). 
The crescent-shaped figure "R G P Q is not shaded in the 
present case, to avoid complicating the diagram ; but, as 
before, the breadth of the figure at any radius (say 3, where 
the port-opening was to be five units) represents the port- 
opening when the crank occupies that radius. We may now 
dismiss the main valve for the present. 

From the general centre O, with radius C X (Diagram 38s 
describe the expansion travel circle y z x ; and at right angle) 
to the radius O ^ as found by Case XXXVIII., draw the 
diameter y x ot the travel circle. Upon O y and O a: as 
diameters describe the two expansion steam circles y o^ o^ and 
X o^ oS and finally, from the general centre O, with radius of 
three units (the slot width), describe the slot circle o^ o^ o^ o*. 

This is all that is required to enable us to trace the slot- 
opening due to both slots throughout the period of steam 
admission. 

The crescent-shaped figure o^ M o^ O, which includes the 
whole amount of horizontal shading (fine and coarse together), 
denotes the slot-opening controlled by the first or outer sloe 
(that nearest the end of the valve), and is simply a copy, upon 
a reduced scale, of the same figure in Diagram 37. As 
explained in Case XXXVII , the amount of opening at any 
given position of the crank is indicated by the radial breadth 
of the shaded portion at the given crank position. 

It is evident from the merest inspection of Diagram 39 that 
slot-opening commences when the crank is in the position 
indicated by the radius O o^, and that the slot is fully open 
upon O ^1 where the radial breadth measures three units. 
This full slot- opening is maintained for exactly half a 
revolution, until the crank arrives at the position O z. Here 
the plate begins to close the slot, so that when the crank is 
upon O 3, the radial breadth scales 2^ units — the prescribed 
amount ; while the shading disappears entirely at the crank 
position O 4^, showing that the slot closes at that point in the 
revolution, giving the -^ths cut-off desired. 

A moment's consideration of the double-slotted valve and 
plate shown in figure 2 will make it clear that while the outer 
slot in the valve, as we have just seen, when once uncovered 
by the movement of the expansion plate, remains clear and 
unobstructed until the return of the plate closes it again at 
the appointed period of the stroke, the second or inner slot, 



100 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

by the passing over it of the bar A, must at a certain point be 
obstructed or closed entirely, and again reopened some time 
before the actual cut-off point is reached. 

It becomes necessary, therefore, to exercise care in pro- 
portioning the valve and plate so as to either keep this 
obstruction within due limits or remove it to a part of the 
stroke where its presence is unimportant. 

Turning now to the consideration of the second or inner 
slot, the extent of whose opening is denoted by the fine 
horizontal lines only, we see that it, like the outer slot just 
mentioned, opens when the crank is upon O o^, and becomes 




—0- ry- 



aCALE OP ikl^lTS 
Diagram 39a, 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 101 

fully open at O z^. From this point, however, owing to the 
obstruction caused by the bar A of the expansion plate, which 
here begins its passage over the slot, the opening is gradually 
reduced, and is closed entirely while the crank is passing from 
the position O o^ to the position O o^, or O H, Here it begins 
to open again, and attains its full opening of three units when 
the crank is at O z. 

After this it closes exactly as the other slot does, being open 
to the prescribed amount, 2^ units, when the crank is upon 
O 3, and closing entirely at the given cut-off point O 4^. 

Now as the main valve does not open the port until the 
crank is upon E (when lead or preadmission begins), we 
need not concern ourselves further with what occurs earlier 
in the stroke, but may proceed at once to collect and transfer 
to the exterior of the four-unit main lap circle R Q P the 
radial breadths of shaded slot-opening, as fully explained in 
Case XXXVII., so as to make up a figure which will show us 
the actual steamway due to the combined action of main 
valve and expansion plate through the whole period of 
admission. 

This is represented by the heavy black lines in the diagram 
(the outer edges being, of course, the real boundary of the 
figure). By its aid it is easy for us to follow the cycle of 
operations, thus : — 

Admission begins when the crank is upon the radius O E, 
by the main valve opening the port in the cylinder face. 
When the crank is upon O K, the steamway is three units 
through both port and slot, as measured by the radial breadth 
K^ K, and although the 2)ort continues to open (attaining its 
maximum opening Q C when the crack is upon O C), yet as 
there is only one slot available, the steamway remains three 
units only until the crank arrives at the position O H, at 
which point the second or inner slot begins to open. The 
steamway now increases rapidly by the opening of the second 
slot, until, with the crank at O G, the steamway is once more 
controlled by the amount, G^ G, of ^"r^opening at this point. 
In other words, here, and while the crank is passing over the 
arc G 3, the slot-opening is in excess of the pore-opening. 
With the arrival of the crank at the position O 3, where, by 
the original conditions, ports and slots were both to be five 
units open, control is resumed by the expansion plate, and a 
rapid lessening of the steamway occurs, terminating, of course, 
with the cut-off, or entire slot closure, when the crank 
reaches 4^. 



102 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

The slot-opening due to the outer* slot alone is denoted by 
the figure (partly in dotted lines) I ^ K P. (Were it not for 
the obstruction which prevents the second slot from opening 
until O H is reached by the crank, Diagram 37 would be 
correct for the two three-unit slots, as it stands). 

We have now to see whether the bar length A, as found by 
construction in Diagram 38, is confirmed by the independen 
evidence of Diagram 39. We stated in case XXXVIII. that a 
bar-length equal to the radius of an arc N N^, struck from 
centre z, touching the main cut-off radius O D, would cause 




SCALE OH UNITS 
Diagram 39b. 



Represented by the inner slot figure in the diagram. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 103 

the outer slot to be reopened simultaneously with the closing 
of the port by the main valve, and we see by the present 
Diagram 39 that the arc N N^ of the same radius struck from 
the centre O cuts the intersection of the expansion circle y 
0^ o^, by the radius O D. 

A second arc L L^, of ^in. greater radius, designed to give 
a margin of safety of that amount, is reproduced as the arc 
L Li in Diagram 39, and shows the reopening (denoted by the 
shading) safely postponed until the crank arrives at O F. 

Perpen(^iculars dropped from L, from O, and from M, in 
Diagram 39, construct the expansion plate full size without 
further trouble, and from it the slots b c, bar ab, or A, and 
distance a c, of the main valve, follow as a matter of course. 

All these results, it will be noted, are derived from the 
dimensions and proportions found, for the given conditions, 
by the simple constructional Diagram 38 ; the intention of the 
present more elaborate figure being mainly as a substitute for 
a full-sized model, whereby every movement of valve and 
plate can be followed and criticised throughout the revolution. 
Whether the steamway we have just been tracing is 
sufficient for the purpose of maintaining the full pressure upon 
the piston up to the point of cnt-off is a question outside our 
present purpose ; the increase in the steamway may or may 
not keep pace with the increasing v^elocity of the piston. But 
we will assume that in the present instance it is insufficient, 
and that a wider or more rapidly-increasing steamway is 
desirable. 

One obvious method is to make the slots wider ; and 
Diagram 39a, to which the same reference letters apply, 
shows the effect of an addition of half-a-unit to the width of 
each slot in valve and plate (the throw and angle of the ex- 
pansion eccentric being modified a little to suit — see Diagram 
38a). It will be at once seen that the crank position O H, 
where the second or inner slot begins to take steam, occurs 
somewhat earlier in the revolution. 

By comparing the black-lined figures representing the 
steamway respectively in Diagrams 39 and 39a, the effect of 
the increased opening is immediately perceptible. The 
steamway is manifestly much more nearly equal to the port- 
opening throughout the admission period ; but at the same 
time the slot-opening is needlessly in excess of the port- 
opening when the crank is at O J. This crank position O J — 
the point at which the obstruction of the second slot ceases— 
is important enough to merit a little consideration on its own 
account. 



104 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

In a valve and plate such as we have been describing, 
where the slots in both are of equal width, the position O J 
indicates the maximum of slot-opening, because at the same 
moment the obstruction ceases, closure commences. This 
maximum slot-opening, in order to get the best effect from 
the combination, should obviously be placed with some regard 
to the point of maximum attainable _2:>or^opening, as deter- 
mined by the action of the main valve. 

The maximum port -opening always takes place when the 
crank is upon O C ; and when the cut-off point is later than 
this in the revolution, we shall evidently be securing the best 
possible steamway under the given conditions by making O J 
coincide with O C. 

This, it may be remarked in passing, has nothing to do 
with the closiug of the slots at a given position of the crank 
(in this case they were to be opened five units at O 3 and 
closed at O 4^), which is a function of the throw and angle of 
the expansion eccentric O x. 

Supposing, therefore, thnt in Diagram 39, O J occurs too 
late in the stroke, and that we wish, while retaining the 
three-unit slots in the valve, and the throw and angle A O X 
of the expansion eccentric (Diagram 38), to get a wide-open 
second slot when the crank is at the point of maximum port- 
opening O C. 

Diagram 39b is an exact copy of Diagram 39, with the single 
exception that the second slot (represented, as before, by the 
fiuely-shaded portions of the figure), is shown as radially dis- 
tant from the centre O an amount equal to the negative 
folloiving lap h n of the slot in the expansion plate. The 
inner arcs V S and V^ S^ bounding this slot in Diagram 39b 
are struck from centre O with the radius of the arc S H from 
centre z in Diagram 38b, and the outer arcs U o^ and U^ o^ 
with the radius of the arc S H in the same diagram. 

The radial breadths of shaded slot-opening are collected and 
transferred to the exterior of the four-unit main lap circle 
precisely as explained in Diagram 39, to make up the figure 
enclosed by the heavy black lines, the slot opening due to the 
first or outer slot''' alone being, asbefore, denoted by the figure 
partly in dotted lines I ^ K P. 

Perpendiculars dropped from L, from O, from M, and from 
T in Diagram 39b, construct the expansion plate as before, 
the distance O M in the diagram (=1 n in the expansion 
plate) being the negative lap of the bacJc {ov following) edge b 

* The iruier slot figure in the diagram. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 105 

of the slot b c. This must not be confounded with the 
negative or minus lap of the cut-off edge of the expansion 
plate, which is here indicated by the distance marked a . 

It will now be seen that with the construction indicated by 
Diagram 38b, the slot-opening fulfils, as demonstrated by the 
present diagram, the required conditions. 

The second slot attains its full opening when the crank is 
on O C, and remains wide open until the crank reaches O J^, 
thus giving a largely increased steam way as compared with 
Diagram 89, where the same eccentric and the same width of 
slot in the valve were employed, but without the negative 
following lap 6 n in the plate. 

The length of bar a n in the valve is, as will be seen, equal 
to the length of bar in the plate, plus the negative following 
lap, OT ah + h n, which again is equal to the distance L M on 
the base line of the diagram. 

The distance a c in the valve (= L T in the diagram) must 
of course coincide with a c in the plate, so that both slots are 
closed at the same moment. 

The Diagrams 39, 39a, and 39b can of course be completed, 
as regards the exhaust portion, from Cases XVI., XVII., or 
XXI. 



Case XL. — Variable expansion by altering the angle of 

THE eccentric, ITS THROW BEING FIXED. PARTICULARS OF 
MAIN VALVE AND ITS ECCENTRIC BEING GIVEN, TOGETHER WITH 
THE MAXIMUM TRAVEL PERMISSIBLE FOR THE EXPANSION PLATE ;'" 
TO FIND THE RESPECTIVE ANGLES OF THE EXPANSION ECCENTRIC 
FOR CUTTING OFF AT GIVEN POSITIONS OF THE CRANK OR OF 
THE PISTON. 

In the diagram let O C represent the throw, and A O C the 
angle with the crank, of the given main eccentric, and let O 5, 
O 4, O 3, O 2, and O 1 be the specified points of cut-off 
reckoned from B, the assumed starting point of the crank (the 
real direction of rotation being indicated by the arrow). 

It is required to find the angles for the adjustable expansion 
eccentric, its throw being limited to a radius of the arc X^... 
..,X\ the expansion plate being of the Series C type, and the 

valve single- slotted (Fig. 1, page 87,). 

* I.e., as limited by the inside length of the steam-chest, or by other 
conditions. 



106 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




Diagram 40. 

With centre C, and radius C a , equal to the minus lap of 
the expansioQ plate, describe the arc of the expansion lap 
circle S a Y. Touching this arc and perpendicular to the 
crank position O 5, one of the specified points of cut-off, draw 
the director Z^ and produce it to cut the given arc X^ X}. 

Its point of intersection X^ is the position of the centre of 
the expansion eccentric, and A O X^ its angle with the crank, 
for cutting off the steam when the crank occupies the radius 
O 5. In like manner the directors Z^, Z^, Z^, Z^, drawn 
touching the arc S a Y, perpendicular to the respective given 
crank positions 4,03,02,01, cut the given arc at X^ X^, 
X^, X^, and these intersections denote the positions, and 
A O X4, A O X\ A O X2, A O Xi, the angles with the crank, of 
the adjustable expansion eccentric for cutting off the steam 
when the crank occupies the respective given positions. 

It is thus an extremely simple matter to construct the diagram 
for variable expansion, the centre of the expansion eccentric 
being adjustable between the limits of X^ as its earliest 
position and X^ as its latest. 

In all cases where the expansion plate is constructed to cut 
off with its outer edges, as in the present case, the expansion 
eccentric leads— i.e., it is in advance of the main eccentric 
OC. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



107 



Case XLI. — Variable expansion by altering the angle 

OF THE eccentric, ITS THROW BEING FIXED. CONDITIONS AS 

IN Case XL. To construct the Zeuner Diagram, and to 

DELINEATE THE CURVES OF PORT-OPENING (STEAMWAY) FOR THE 
several POINTS OF CUT-OFF. 

In Diagram 41 the throw and acgle A O C of the roain 
eccentric, and the throw and angular positions of the expansion 
eccentric for the given points of cut-off, are those given or 
determined by Case XL. 



A- 




Diagram 41. 

The lap of the main valve is 4 units and the lead is 1 unit ; 
the maximum permissible throw of the expansion eccentric is 
15 units ; the given points of cut-off are fths, ^, fths, ^th, and 
^th of the piston stroke (reckoned from the assumed starting 
point B), the valve has a single slot of 6 units width, and the 
expansion plate has a minus lap (a Fig. 1) of the same 
amount, 6 units. It is required to construct a diagram show- 
ing the exact amount of steamway open (for each position of 



108 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

the expansion eccentric) at any crank position from O up to 
the point of cut-ofl'. 

With centre O, and radius O C, describe the main travel 
circle A D C E, and upon O C as a diameter describe the main 
steam circle O R C P. Again, from centre O, with radius O P, 
equal to the given lap of main valve (4 units), describe the 
main lap circle L R P. The intersections of these two latter 
circles, at P and R, mark the opening and closing of the main 
valve (Case XVI.) ; and the radii O E and O D, passing 
through P and R, denote the position of the crank at each 
operation respectively. 

Now join X5, C X^ C Xi (Case XL.), and draw O y\ 

O 7/4 j/i, equal and parallel each to each ; and similarly, 

upon the opposite side of the general centre 0, make O x^, 
O x^ O x\ also equal and parallel to C X^ C X^ C X^. 

O x^, O x^ O x^, will be the throws, and A O ;\;^ A O x^... 

...A O x'^, the angles, with respect to the real position of the 
crank at A, of eccentrics which would fulfil the respective 
given conditions as to cut-off if the main slide valve were 
fixed, forming a stationary face for the expansion plate to 
work upon. 

Upon O y^, O y'^ O y'^, as diameters, describe their 

respective expansion circles ; f and from centre O, with radius 
equal to the slot width (6 units), describe the slot circle 
N o5 oiM. 

Each of the expansion circles will cut the slot circle at the 
point where the latter is intersected by the corresponding 

crank radius — viz., at the points o^, o^ o^ ; and the arcs 

O 0'^ O 0^ O o^, within the slot circle, will give, if trans- 
ferred to the exterior of the main lap circle L R P, as fully 
explained in Case XXXVII., the curves of steamway (dis- 
tinguished by the thick black lines), showing, for each given 
adjustment of the expansion eccentric, the exact amount of 
slot-opening from the crank position O E, where admission 
begins, up to the point of cut-off. 

in this way the action of different valves and valve gears 
can be readily compared, and the most suitable for the 
requirements selected, with an accuracy which leaves nothing 
to be desired, and with the least possible expenditure of time 
and trouble. 

In the present case we have a single- slot valve, and no 
question of reopening can arise ; but, as noticed in Case 
XXXVIII. with a douhle-^XoiiQdi valve having an expansion 
plate of variable travel, care must be taken that the slots are 

t In the diagram semi-circle, to avoid complexity. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 109 

SO far apart in the valve that at no position of the eccentric 
can a reopening occur. 

Assuming for the moment that there were two slots in the 
valve in the present case, their minimum distance apart — i e., 
the length of the bar A, Fig. 2 — would have to be equal to the 
diameter O y^ of the largest expansion circle in Diagram 41. 
If, as in Case XXXIX., we simply produce the crank radius 
O D (marking the position of the crank at the moment the 
main valve closes the port) to cut the expansion circle at a, 
and made O a the bar length, we should have a reopeniog 
when the crank arrived at O a} (as denoted by the shaded arc 
extending from O (X^ to O a), and the slot would close again 
when the crank occupied the position O a. 

With a bar length O y'^, the slot would barely close when 
the crank was in the position denoted by O y^. 

But by making the bar length A equal to 6 6, which is, say, 
^in. in excess of O y'^, we secure a margin of safety of that 
amount when the expansion plate is at the extremity of its 
greatest travel, J and thus no reopening can occur at any part 
of the stroke in any position of the adjustable eccentric 
between the given limits. 

The diagram may be completed, as regards the exhaust 
portion of it, from Cases XVI., XVII., or XXI. 

Case XLII. — Variable expansion by altering the throw 
OF the eccentric, its angle being fixed. Particulars of 
main valve and its eccentric being given, together with 
the maximum travel permissible for the expansion plate ; 
TO find the respective throws of the expansion eccentric 

FOR cutting off AT GIVEN POSITIONS OF THE CRANK OR OF THE 

piston. 

In Diagram 42, the throw O C and the angle with the crank 
A O C of the main eccentric, are assumed identical with those 
employed in Case XL. It is required to find the throws of 
the variable expansion eccentric, its maximum throw being 
in the present case limited to 15 units ; the expansion plate 
being of the Series C Type, and the valve single-slotted. 
(See Fig. 1.) 

Let O 5, O 4, O 3, O 2, O 1, and O O- be the specified points 
of cut off, reckoned from B, the assumed starting point of the 
crank (the real direction of rotation being indicated by the 
arrow) . 

With centre C and radius C a equal to the minus lap of 
the expansion pla/te, describe the arc of the expansion lap 

X Greatest travel — see footnote in latter part of Case XXXV. 



110 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




SCALE or UN/TS 



DIAGRA.IVI 42. 

circle S a Y; and with centre O and radius equal to 15 units, 
the specified maximum throw of the expansion eccointric, 
describe a short arc a b. At right angles to O 5, the position 
of the crank where the latest desired point^ of cut-off is to 
occur, and touching the arc S a Y, draw the director Z^, and 
produce it to cut the short arc a b. Its point of intersection 
X^ is the position of the centre of the expansion eccentric for 
the specified latest point of cut-off. 

Join O X^ then A O X^ will be the constant angle of the 
expansion eccentric (referred to the real position of the crank 
at A) for all points of cut-off. 

Now, at right angles to the respective crank positions O 4, 
O 3, O 2, O 1, and O O (i.e., O B, the assumed position of the 
crank at the beginning of the stroke), draw the directors Z^ 

Zo, all touching the arc S a Y, and produce each of them 

to cut the radius O X^. (The director Z° will cut O X^ pro- 
duced, to the left of the centre O, but we shall come to that 
presently.) 

The points of intersection with O X^, and X^ produced — 

viz., X^ X°, — are the successive throws of the variable 

expansion eccentric for cutting off at the respective points 
specified ; the eccentric (either actually or virtually by meaos 
of a link gear) being movable at the constant angle A O X^ 
between the limits of X° as its earliest position, and X^ as its 
latest. 

Now, in cases where a cut-off at O of the stroke is necessary, 
as it usually is with automatic expansion gear, it is not an 

* I.e., cut-off at zero for automatic expansion gear. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. Ill 

UDCommon thing to find that the designer has omitted to 
make provision in his Unk motion for the very short minimum 
travel required. The effect of this is that when the pair of 
eccentrics are fixed upon the crank shaft in their assigned 
positions, the earhest attainable cut-off is perhaps ^th or 
i'sth of the stroke, and the governor is therefore unable to 
control the engine effectually. 

The erector's private remedy in such cases is to move the 
pair of eccentrics together further round the shaft in tie 
direction in which the engine is to run, until the expansion 
plate ivill cut off at zero, or near it ; thus increasing the lead 
of the main valve, and making every operation in the cycle 
occur earlier than was intended. The power of an engine 
may be very much reduced in this way, and I>iagram 42, in 
which this evil is purposely exaggerated, is given as an 
example to be avoided in designing automatic expansion gear 
of the pivoted-link class, where only one eccentric is employed 
to move the expansion plate. It will be seen that the throw 

X° of the expansion eccentric, in order to give a cut-off at 
zero, is less than nothing— -i.e., it is negative, &^ just now 
noticed ; and this is a consequence of the distance O P (or 
lap + lead of the main valve) being less than the distance 
C S (or C A, the minus lap of the expansion plate). 

To cause the director Z*^ (which is a tangent to the arc 
S A Y, perpendicular to the base line of the diagram) to fall 
to the right of the centre O, either the lap or the lead, or 
both, must be increased, or the minus lap of the expansion 
plate reduced. Diagram 44 shows the minimum throw O X*^ 
of the expansion eccentric, with a positive length of a little over 
two units, by the adoption of a double-slotted valve, and con- 
sequent reduction of the minus lap a by one-half. 

Case XLIII. — Variable expansion by altering the throw 
OF the eccentric, its angle being fixed. Conditions as 
in Case XLII. To construct the Zeuner Diagram, and 
TO delineate the curves of port- opening (steamway) for 
the several points of cut-off. 

In Diagram 43, the throw and angle A O C of the main 
eccentric, and the angular position and throws of the expan- 
sion eccentric for the given points of cut-off, are those given 
or determined by Case XLII. 

The lap of the main valve is, as before, 4 units, the lead is 

1 unit, and the port width is, of course, 6 units. 

The maximum permissible throw of the expansion eccentric 
is 15 units; the given points of cut-off are fths, ^, fths, :^th, 
|th, and O, of the piston's stroke (reckoned from the assumed 



112 THE PROPORTIONS AND MOVEMENT OF SLIDB VALVES. 




SCALE OF UNITS 



Diagram 43. 

starting point B), the valve has a single slot of 6 units width, 
and the expansion plate has a minus lap ( a , Fig. 1) of the 
same amount, 6 units. 

It is required to construct a diagram showing the exact 
amount of steamway open (for each separate throw of the 
expansion eccentric) at any crank position from O up to the 
point of cut-off. 

With centre O and radius O C, describe the main travel circle 
A D C E, and upon O C as a diameter describe the main steam 
circle O R C P. Again, from centre O, with radius O P equal 
to the given lap of main valve (4 units), describe the main lap 
circle L R P. The intersections of these two latter circles at 
P and R mark the opening and closing of the main valve 
(Case XVI.), and the radii O E and O I) passing through P 
and R denote the position of the crank at each operation 
respectively. 

Now join C X^ C X^ C X^ (Case XLIL), and draw 0:v^ 

O y^ O jy° equal and parallel each to each ; and, similarly, 

upon the opposite side of the general centre O make O x^^ 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 113 

O x^ O A'O, also equal and parallel to C X■^ C X^ C X^. 

O ;ir5, O AT^ O x'^ will be the throws, and A O x\ A O x^ 

A O Ar° the angles, with respect to the real position of the 
crank at A of eccentrics which would fulfil the respective 
given conditions as to cutoff, if the main slide valve were 
lixed, forming a stationary face for the expaasioa plate to 
work upoi. 

Upon O )'5, O yi O y^ as diameters, describe thpir respec- 
tive expausioQ circles; and from centre O, with ra'lias equal 
to the slot width (6 units), describe the slot circle N O^ O'^. 

Each of the expansion circles will cut the slot circle at the 
point where the latter is intersected by th*^ corresponding 

crank radius— viz., at the points 0% O'' 0° ; and the arcs 

O o^, O o^ O o", within the slot circle, will give, if trans- 
ferred to the exterior of the maiu lap circle L E, P, as fully 
explained in Case XXXVII., the curves of steamway (distiu- 
guished by the thick black lines), showiog for each given 
throw at the constant aogle of the expansion eccentric, the 
exact amount of slot opening from the crank position O E, 
where admission begins, up to the poiot of cut-off. 

As in Case XLI., the minimum length of the bar A in the 
expansion plate (assuming for the moment that we are using 
a double-slotted valve) would have to be made equal to the; 
diameter O y^ of the largest expansion circle (in Diagram 43)^ 
l)lus whatever margin of safety (say ^in ) we may consider 
desirable, so that no reopening can occur at any part of the 
stroke at any throw of the constant-angle eccentric between 
the given limits. 

By comparing the curves in the present Diagram 43 with 
thob-e of Diagram 41, the respective steamways obtained by 
the two alternative methods of producing variable expansion 
can be readily contrasted, as the proportions are identical in 
the two cases. 

The diagram should be completed, as regards the exhaust 
portion of it, from Cases XVI., XVII., or XXI. 

Case XLIV. — Variable expansion by altering the throw 

OF THE expansion ECCENTRIC, ITS ANGLE BEING FIXED. To 
CONSTRUCT THE VALVES AND ECCENTRICS FOR A GIVEN CYLINDER 
FACE TO FULFIL CONDITIONS AS SPECIFIED, THE VALVE BEING 
DOUBLE OR MULTIPLE SLOTTED, AND THE EXPANSION PLATE OF 

THE Series C type. 

(1.) The present case, which is mainly a summary or 
recapitulation of portions of previous cases, is intended to be 
complete in itself, as a guide to the whole process of getting 
out a set of valves and eccentrics from the given data to fulfil 



Hi THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

certain specified conditions. The diagram employed is of the 
simples J possible character, and the method of Zeuner is 
purposely avoided, with a view of reducing the problem to 
elementary simplicity. 

First of all there are one or two poiats which may be briefly 
abstracted from the introductory chapter of this series.^ 

(2.) Beal mid Assumed Positions of the Cranh. — The real 
position of the crank m all the diagrams is to the left hand of 
the reader, and will be always denoced by the letter A ; and 
the angular positions of the eccentrics, as shown in the 
figures or found by construction from the given data, are 
always with reference to this real position of the crauk at A. 
But for all other purposes of reference the crank is to be sup- 
posed to start from the poiat B, upon the right-hand side of 
the diagram, or diametrically opposite to its real position ; 
the arrow in all cases denoting the real direction of rotation. 

(3.) The diagram is drawn to a scale, the unic of which is, 
not any fraction of an inch or of a foot, but one-'^ixth the 
'width of steam port in the cylinder face. Thus if the steam 
poit were, say, fin. wide, the unit would be ^in. 

(4.) For the sake of brevity in description the word valve 
is only applied to the main valve, the cut-off valve beiog 
termed the expansion plate ^ or the plate simply. Similarly the 
word port is applied only to the ports in the cylinder face ; 
the portis through the valve and in the expansion plate being 
called slots. Thus the valve opens and closes ports in the 
face of the cyliuder, and the plate performs the same func- 
tions with regard to the slots in the valve. | 

(5.) Lastly, in practice the diagrams should be drawn to 
as large a scale as possible (at least double the actual size), 
for the sake of clearness and accuracy in construction. 

(6.) Tne valve aud plate being understood to be of the 
Series C type (see Fig 3), the given conditions may be 
expressed as follows : — 

Cylinder face : Steam-port width 6 units. 

. , Exhaust port width 12 

,, Bridges or bars width 6 

Main valve : Lead 1 

,, Port opening atf of tlie stroke 5 

,, Exhaust lapt 

Width of the slot (2 slots) 3 

Expansion plate : Limits of cut-off, and jfeths of the stroke. 

Expansion plate : Slot-opening at fths of the stroke when the eccentric is at 

maximum throw, 5 units, or 2^ units each slot. 
Expansion plate : Minus or negative lap ( A , Fig. 3), 3 units. 
Expansion eccentric: Minimum throw— i.e., when the plate cuts off at O of 

the stroke - not less than 2 units. 

* Pages 5, 6 and 7. % Page 39. 

t I.e., the cavity in the valve just spans the bridges or bars. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 115 

With these conditions before us, we may proceed to con- 
struct Diaeram 44 as foUows: — 




SiVALB OF UNJTS. 

Diagram 44. 

(7.) In any strai^^ht line A B take a point O, and lay off to 
the right of it the distance O Z*^, equal to the least allowaV-le 
throw of the expansion eccentric (2 units). From O TI^ lay 
off, still to the right. Z° Q, equal to the minus lap of the 
eKpansion plate a, Fiij. 3 (here 3 units). At Q erect a per- 
pendicular, the lead-line. The centre C of the main eccentric 
will be somewhere on this line. 

(8.) As the distance O Q is lap + lead (Case I.), lay off 
upon the line A B to the left of Q the distance Q P equal to 
the given lead % (1 unit). Then O P will be the lap of the 
main valve. From centre O with radius O P describe the lap 
circle L R P. 



Learf, the amount the port is open at the beginning of the piston stroke. 



116 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

(9.) Now let O 3 represent the position of the crank (sup- 
posed to start from B) when the piston is at fths of its 
stroke, § and the port is to be open the given amount (5 units). 
Lay olf (Case VII.) upon tbe hne O 3 the distance S T 
exterior to the lap circle, equal to the given port-opening (5 
units). 

(10.) From the point T draw the line T C at right angles 
to O T, cutting the perpendicular Q C at C. Join O C, which 
will be the throw, and A O C will be the angular position of 
the main eccentric with respect to the real position of the 
crank at A With centre O and radius O C, describe the 
main travel circle A C B F. 

(11.) Touching the lap circle L KP, and at right angles to 
the radius O C, draw the chord D E of the travel circle 
A C B F. Join E (which will be the position of the crank 
when admission takes place), and D (which will be its posi- 
tion when the main valve closes the port). 

(12.) Ordinates let fall from any of the points (as D, or 3, 
or E) in the travel circle, upon the base line A B, will denote 
the fraction of the jJ'i'Ston's stroke (reckoned from B) at which 
the respective operations occur. 

(13.) Now (Case XXXVIII.), with centre C and radius C a 
equal to the minus lap of the expansion plate (3 units), 
describe the arc of the expansion lap circle Y a Y^. Touching 
this arc, and perpendicular to the given crank positioti O 4^ 
(the prescribed viaximum point of cut-off), draw the direc'or\\ 
Z and produce it to X or further. Upon the radius O C lay 
off the distance a a^ equal to half (for a ^M;o-slotted valve) 
the desired total slot-opening (here supposed 5 units when 
the crank is on O 3, hence the distance a a^ will be 2^ 
units); and with centre C and radius C a^ (here half-a-unii) 
describe a small arc. 

Touching this small arc, and perpendicular to the crank 
position O 3 (where each slot is to be open the prescribed 
amount a a^ or 2|^ units), draw the director and produce it 
to X, the point of intersection with the director previously 
drawn. This point X will be centre of the expansion eccentric 
at its maximum throw. Join O X, then A O X will be the 
constant angle with respect to the real position of the crank 
at A, of the expansion eccentric for all points of cut-off to 
fulfil the given conditions. 

§ Neglect the error caused by the angularity of the connecting rod— /.e., 
assume it of .infinite length (pages 5 and 6). 

II Director, a perpendicular to any crank position drawn touching the are 
Y A Y\ and produced to X or further. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES, 117 

(14 ) The director Z° drawn at right angles to the crank 
position O B (corresponding to zero of the piston's stroke) 
will cut X at X'', and O X^ will be the required minimum 
throw of the expansion eccentric (for cuttiog off at O of the 
piston's stroke), which was not to be less than 2 units. 

(15.) We have now all the angles and dimensions fixed to 
fulfil the required conditions as to steamway^ at the given 
crank position O 3 ; and as to cut-off at the two extremes — 
VIZ., the crank positions O B and O 4|^, corresponding respec- 
tively to and -^ths of the piston's stroke. The throws of 
the constant-angle eccentric for cut-offs at earlier positions of 
the crank, as O 3, O 2, and O 1, are indicated upon the lioe 
O X by the poiots of intersection X^, X^, and X^ of directors 
Z"\ Z^ and Z\ drawn perpendicular to the respective crank 
positions, touching the arc Y a Y\ 

(16.) But it may be that the maximum throw O X of the 
expansion eccentric (here about 13 units), as found by the 
foregoing process, is inadmissible, owing to the interior length 
of the steam chest or the maximum obtainable movement of 
the link being insufiicient. la such a case, we should have to 
be content with a reduced steamway at the given crank posi- 
tion ; and the sinaplest plan of procedure is to try the effect 
of reducing S T by, say, half-a-uuit, and, from the new centre 
C thus found, describing a new arc Y a Y^ (still of 3 units 
radius), and a new small arc whose radius C a^ is increased 
by one-quarter='' of a unit. The clear steamway through the 
two slots and the port would then be 4^ units instead of 5, 
and a shorter full throw, O X, of the expansion eccentric would 
be indicated by the intersection of the new directors, leaving 
the prescribed lead and point of cut-off unaffected. 

(17.) This being satisfactorily adjusted, we may proceed to 
find the har length, or the interval of solid plate between the 
end of the expansion plate and the edge of the slot (A, Fig. 3). 

Join C X and C X°, and note which of these distances is 
the greater. In the present case C X" is the greater. C X*^ 
therefore denotes the greatest movement of the expansion 
plate relatively to the main valve, upon the back of which it 
slides. If we made the bar A (Fig. 3) equal in length to 
the distance C X^, it would jast barely prevent the slot 
in the plate uncovering the outer slot in the valve during 
the period of expansion. Bub by making the bar length 

^ I.e., the clear opening through both slots and port ;here 5 units). 
* Thus reducing the opening of each slot by that amount. 



118 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

A (Fig. 3) say, ^io. in excess of C X*^, we secure a margin 
of Sfifety ot that amount when the expansion plate is 
at the extremity of its greatest travel (relatively to the main 
valve), and thus no reopening can occur at any part of ihe 
stroke at any throw of the constant-angle eccentric between 
the given limits. 

Add, therefore, ^in. (or whatever margin of safety be 
thought necessary) to C X°, aud that will be the correct 
length of the bar A in the expansion plate. 

(18.) We have now to fix the correct amount of negative 
following lap {b n, Fig. 3) ia the expansion plate. The effect 
of this widening of the slot is to cause an earlier opening 
of the second or inner slot in the valve than would otherwisn 
be the case. This question has been fully discussed and 
illustrated in Cases XXXVIII. and XXXIX., and we may 
proceed to apply it here with no more recapitulation than js 
absolutely necessary. It may be noted that it is in general 
only with reference to the maximum point of cut-off (here at 
^ths of the stroke) that the delayed opening of the second 
or inner slot in the valve (due to the passing over it of the 
bar A) need be considered at all, because the obstruction 
rapidly lessens as the admission period is reduced. 

Taking, therefore, the maximum throw of the expansion 
eccentric, or O X, as a basis, we may ascertain at what point 
in the revolution the second slot begins its opening, as 
follows:— From centre O, with radius C X (not O X), describe 
the circle (or portion of a circle) y z x, and at right angles to 
the line C X connecting the centres of the two eccentrics, 
draw the radius O z. From the point z, where this radius 
cuts the semi-circle y z x, describe an arc S 11°, whose radius 
is A , the width of slot 3 units. The radius O 11°, drawn 
touching this arc, will indicate the position of the crank 
(supposed to start from ti) when the second slot begins to 
open, on the assumption that the slot in the expansion plate is 
3 units wide, the same as the valve slots (see Case XXXVIII. 
and Fig. 2). The radius O ^ marks the position of the crank 
when the expansion plate is at the centre of its travel on the 
back of the main valve, and both slots are for the moment 
wide open. Immediately after this slot- closing commences. 

Now we can cause the second slot to begin its opening at 
any point in the revolution earlier than O Ho by iucreasing 
the width of the slot in the expansion plate, as in Fig. 3 (and 
Diagram 38b). Thus, suppose we wish the slot to attain its 
full opening when the crank occupies the position O C, for 
example (it will still not begin to close until the crank reaches 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 119 

O z), we make the extra width of the slot equal to tlie radius 
of the arc b n, struck from centre z to touch the desired crank 
position O C. The total width of the slot {i.e., b n -\- S units, 
or b c, Fig 3) is shown in the diagram as an arc b c, struck 
from centre z with radius 3 units greater than arc b n, aud 
the crank position O H'^ drawn to touch it indicates the 
earlier position of the crank when the second slot begins to 
open This point z is the key of all the operations effected 
by the movement of the expansion plate. For instance, an 
arc U S of radius a (3 units) will touch the crank position 
O 4^, where closing occurs ; and a small arc of half-a-unit 
from the same centre will touch the crank position O 3 where 
the prescribed opening was to be 2^ units for each slot {i.e., 
half-a-unit less than the full openinjj). After z is reached in 
the revolution of the crank slot-closing commences, as already 
noticed. 

(19.) We have one more dimension to fix- the length of 
bar in the valve. This would be of the same length as the 
bar A in the expansion plate, if the slot in the plate were of 
the same width as the slots in the valve (see Fig. 2). But in 
order to suit the widened slot. Fig. 3, the length of bar in the 
valve has to be increased by the same amount b n. Hence 
the valve-bar length is expressed as A + 6 n (Fig. 3). 

(20.) By the given conditions there is to be no exhaust lap, 
but we may (Case XII.) show on the diagram the points of 
release and closure ot the exhaust. 

In Diagram 44 produce the radius O C to F. Draw the dia- 
meter G G^ at right angles to the diameter C F, and the 
radius O G will represent the position of the crank when 
release occurs, and the radius O G^ will denote its position 
when closure commences. 

(21). To determine the portion of the stroke during which 
the exhausting port remains wide open.''' Lay off upon O F 
the distance O V, equal to the width of the steam port 
(6 units), and draw the chord J K parallel to G G^. Join O J 
and O K. The port remains wide open to the exhaust, while 
the crank revolves through the arc JFK. From G to J the 
port is cpening, and from K to G^ it is closing. 

Ordinates drawn from any of these points to the line A B 
will denote the fraction of i\ie piston's stroke (reckoned from 
the assumed starting point B) at which the respective opera- 
tions occur. 

* Not the exhaust port, but the steam port, which for the moment is 
exhausting. 



120 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




\2 18 24 SO 36 — 42 48 — M- 

SCALE OF VNTTS 



Fig. 4, Series C. 



(22.) Fig. 4 is a reproduction of Fig. 3, with the addition 
only of some reference letters connecting the dimensions of 
the expans on plate, main valve, and cylinder face, with the 
corresponding lines on Diagram 44, as shown in the following 
table : — 



Description : 

Valve, Plate, and 

Cylinder Face 




References tr 




Fig. 4 


Diagram 44. 


Paragraph, 
Case XLIV. 


Expansion plate- 
minus lap 


A 

A or a. & 
bn 
b c 
B 

(CorA+&») 

d 

D 

E 

None 

F 

G 

6 units 

12 units 

H 


Oa 

C X° + lin. 

Radius of arc b n. 

Radius of arc b c 

Ad libitum, hut not less 

C X° + iin. +b 71 

3 units 

6 units 

P 

None 

PQ 

/24 units, or exhaust 

\port plus both bars 

6 units 

OV 

(Twice the 
(Same as 


6 


bar length 


17 


\\egd.t\\Q following lap 

total width of slot 

length of solid part . . 
Main valve- 
bar length 


18 

18 

thanACorCX^+lin.) 

19 


slot widtn 


6 


slot vfidth in under face 
lap 


Same as steam port 

8 




6, 20 


lead 


6 


length of cavity 

height of cavity 

Cylinder face- 
width of steam port 
width of exhaust port 
width of bars 


Same as steam port 

6 
steam port) 
steam port) 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES, 121 




'M- — 36 42__48. 

SCAtt OF UNITS 



FlCx. 5, SERIES C. 

(23.) Fig. 5 is an outline diagram showing a portion of the 
crank, and the main and expansion eccentrics, drawn to the 
same scale as Fig. 4, with references to Diagram 44 and 
Case XLIV. 



References. 



Description : Eccentrics. 


Fig. 5. 


Diagram 
44. 


Paragraph 
Case XLIV. 


^lain eccentric, throw 


C 

AOC 

OX 

OX° 

AOX 

Arrow 


C 
AOC 
OX 
OX° 
AOX 
Arrow 


10 or IC 


,, angle with crank 

Expansion eccentric, maximum throw . . 

,, minimum throw 

angle with crank . . 
Real diiection of rotation 


10 or 16 
13 or 16 

6, 14 
13 or 16 

2 









(24.) We are now in possession of all the dimensions and 
angles necessary to construct a set of valves and eccentrics 
for a variable-throw automatic expansion gear of the Series C 
type. As this type of expansion plate is perhaps the most 
generally useful of the three kinds we have examined, it has 
been treated at somewhat greater length tlidu its predecessors. 



122 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

CHAPTER VI. 

THE MEYER CUT-OFF GEAR. — SeRIES D. 

In any slide valve, or combination of slide valve and ex- 
pansion plate, the fraction of the piston's stroke at which the 
steam port or slot is made to close, is dependent upon three 
principal conditions — namely, the angle of the eccentric in 
relation to the crank ; the throw of the eccentric ; and thirdly, 
the lap of the valve or plate which cuts off the steam ; and 
the point of cut-off is directly affected by any variation in the 
proportions of any one of these determining conditions. 

We have, in preceding sections, considered systems of 
variable expansion gear based upon alteration in the angle cf 
the expansion eccentric, and others in which its throw was 
adjustable for the same purpose. We have now before us the 
discussion of the third system, known as the Meyer cut-off 
gear, in which both the angle and the throw are fixed, or in- 
variable, and the different points of cut-off required are 
determined by an alteration in the lap of the expansion 
plate. 

For this purpose the expansion plate is made in two 
separate halves, united only by a right and left-handed screw, 
which usually, but not necessarily, is formed upon the valve 
spindle itself. By turning this screw it is obvious that the 
two halves of the expansion plate can be made to approach 
or recede with respect to eacii other, thus shorteoing or 
lengthening the distance between the cut-off edges of the ex- 
pansion plate, and so varying the point of cut-off. 




8-^.24— .30i — 36 — 42— 48— .54 
SCALE OJ^ UNITS 



Valve with Single Slots and Series D Expansion Plate. 
Fig. 1, Series D. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 123 

Fig. 1 illustrates the inside or Series D type of Meyer 
expausion plates, shown in the position in which the two 
halves are supposed to be when at their nearest to each other, 
having then a ^?Zms or positive lap equal to the distance a. 
The main valve is of the same character as that illustrated in 
Fig. 1, Section V. 

The plates, when screwed together (as showc), effect their 
earliest cut off, and when separated to their fullest extent 
admit the steam up to the latest desired point of cut off, the 
plus lap changing to minus as the plates are screwed farther 
apart 

In cases where the variable expansion gear is automatic 
{i.e., controlled by the action of a governor) it is commonly 
advisable to arrange for a zero cutoff ; otherwise, where the 
degree of expanf»ion is regulated by hand (as, for example, in 
the low-pressure slide valves of a compound engine, or where 
a throttle valve is employed), the range may vary from, say. 
^th of the stroke onwards, it being always desirable to avoid 
an excessive amount of in-and-out screwing when changing 
from one degree of expansion to another. 

Being an inside expansion plate, the expansion eccentric is 
set to foUo'W the main eccentric, as in Series B, instead of 
leading. 

Case XLV. — Variable expansion by altering the lap of 

THE expansion PLATE. THE THROW AND ANGLE OF THE 
expansion ECCENTRIC BEING FIXED. To CONSTRUCT THE 

VALVES AND ECCENTRICS FOR A GIVEN CYLINDER FACE TO 
FULFIL CONDITIONS AS SPECIFIED ; THE LAP OF MAIN VALVK, 
THE LEAD, THE EARLIEST AND LATEST POINTS OF CUT-OFF, AND 
THE WIDTH OF TdE SINGLE SLOT BEING GIVEN. 

In the first instance we will take, as the condition to be 
fulfilled, an assumed restriction as to the amount of screw 
movement permissible. 

By this term of course is meant the permissible limits of 
variation for each half of the expansion plates — say ten units. 

We may apportion this as follows — viz., a variation from a 
plus lap of four units (a, Fig. 1) to a minus lap of six units. 
The earlier cut-offs are effected by plus lap, and the later 
ones by minus lap. 

In the present example we assume that the required limits 
of cut- off are ^th and |ths of the piston's stroke, and that the 
width of the single slots in the valve is six units. 

We may now proceed to construct Diagram 45, as follows : — 



124 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



im^^. 




THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 125 

In the straight hne A B, let O be the centre of the crank- 
shaft, and O B the assumed starting point of the crank (its 
real direction of rotation being denoted, as before, by the 
arrow). Let 1 and O 5 represent respectively the positions of 
the crank at ^th and at |ths of the piston's stroke, the earliest 
and latest desired points of cut-off. Lay off, upon O 5, 7J\ 
equal to six units, the prescribed minimum Jap (^.e,, tlje 
extreme minus lap) ; and, perpendicular to O 5, draw the 
director at Z^ producing it to y or farther. 

Upon O 1, produced below the axis O, lay off similarly, but 
downwards, the distance O Z^, equal to four units, the pre- 
scribed extreme plus lap ; and, perpendicular to O 1, draw 
the director at Z^, and produce it to j', the point of intersection 
with the director Tl\ Join O j, which would be the throw 
(and A O 3' would be the angle with respect to the real 
position of the crank at A) of an eccentric which would fill 
the prescribed conditions, with the given laps, if the main 
valve were fixed at the centre of its travel, forming a 
stationary face for the expansion plates to work upon. (We 
may conveniently proceed upon this assumption for the 
present, leaving the main eccentric and the motion of the 
main valve to be given or determined later on in the case). 

Now make O x equal and opposite to O t/, and upon y x, as 
a diameter, describe a semi-circle. At right angles to y x 
draw a radius of this semi-circle, O z. 

O z represents the position of the crank (assumed to start 
from b) where the steam would be cut off with no lap, either 
plus or minus, of the expansion plate- ^.e., when the cut off 
edges a, a (Fig. 1) have by a screw movement of four units 
been brought into coincidence with the inner edges z, z of the 
valve slots, a position identical with that of the fixed ex- 
pansion plate in Series B. 

It is, of course, hardly necessary to point out that in the 
present series, lap, whether plus or minus, is measured from 
these inner or cut-off edges z, z of the valve slots, when the 
valve and the plates are in mid-travel and their centres 
coincident. 

Through the point z, in the semi-circle just found, draw the 
horizontal straight line g g, which will serve to represent the 
upper surface of the main valve {g g, Fig. 1) upon the actual 
scale of the diagram. 

If now we take the point z in the straight line ^ ^ in 
Diagram 45 to represent the inner edge z of the left-hand slot 
in Fig 1, we can construct the left-hand expansion plate (of 



126 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

which the right-hand plate is a reversed facsimile) directly 
from the diagram. 

An arc struck from centre ^, to touch the crank radius O 1, 
will cut the line g g s.t a, and the distance z a will be the lap 
required to effect the ^th cut-off, which was to be four units. 
Another arc struck from centre z, to touch the crank position 
O 5, will cut the line ^ ^ at the point c. The distance z c will 
indicate the minus lap required for the fth-^, or latest desired 
cut-off, which was to be six units. And, similarly, for inter- 
mediate positions of the crank, as O 4, O 3, and O 2, arcs 
struck from z, touching those crank radii, will give, at their 
intersections with the line g g, the required laps for the 
respective points of cutoff— viz., |, f ths, and | of the piston's 
stroke; while, as just now noticed, a lap of 0, or zero, corres- 
ponds with a cut-off when the crank occupies the position O z. 
Laps to the right of the point z are, of course, positive or plus 
laps, and those to the left of z, negative or minus. 

We have now upon the horizontal line g g the laps clearly 
shown for each eighth of the piston's stroke between the 
specified limits of ^th and ^ths, and these are all comprised 
within the given range of screw movement, which was to be 
ten units — viz., four units of plus, and six units of minus lap. 
We have now to find two important dimensions, for it will be 
ijoted that hitherto we have only determined positions or 
screw movements. We want to get the dimensions denoted 
by A and by B, in Fig. 1, both of which require careful 
consideration. 

If the length of the expansion plate A be insufficient, the 
slot will be reopened, or uncovered by the traiHog edge b of 
the plate, before the port in the cylinder fac^. has been closed 
by the main valve. This would cause a readmission of steam 
during the period of expansion. This length A must of course 
be determined with reference to the position of the plates 
when they are placed as shown in Fig. 1 —i.e., at their 
maximum of plus lap. Measuring from the leading edge a of 
the plate, we know a z to be four units, and we know a c to 
be ten units (as the width of slot z c is six units by the given 
conditions). We have therefore to find the distance c h, or 
what may be called the trailing lap of the plate. 

Let O D repre;.ent the position of the crank at the instant 
the main valve closes the port in the cylinder face.''' From 
rue point z describe an arc N^ N, touching this crank radius 
O D. Now, if the distance c h were made equal to the radius 

* We have not fixed this or any other operation of the main valve yet, but 
lor the moment y^e may assuma it to be given or determined. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 127 

of this arc, we should have a reopenin*:; by the expaDsion 
plate at the precise instaut that the main valve had closed the 
port. 

A margin of safety is desirable, and for this purpose ^th of 
ail inch, whether the valve be a large or a small one, is not 
too much. If, therefore, the length c 6 of the expansion plate 
be made ^iu. in excess of tbe length of radius of arc N^ N, the 
desired margin of snfety will be st^cured, and the crank radius 
O F, drawn to touch arc L^ L (which is of ^in. greater radius 
than arc N^ N), will indicate the position of the crank when 
the slot harmlessly reopens. 

We have, then, as the total length A of the expansion plate, 
az + zc + cb, where a- 2 is the extreme plus lap, z c the 
width of slot iu the valve, and c b the radius of an arc struck 
from centre z to touch a crank position O F later in the stroke 
tban the position O D, at which the main valve closes. 

We have now to determine the mini mum distance separating 
the two inner edges z, z of the slots in the valve, the distance 
B, Fig. 1. If this distance B be insufiQcieut, the left-hand 
slot may be partially covered by the right-hand plate when 
the plates are at the extremity of their travel to the left, or 
vice versa. 

The minimum distance B necessary to secure an un- 
obstructed slot is one-half the actual movement of the plate 
upon the back of the valve plus the extreme plus lap of the 
plate, or O J' t + i*^^^ units in the present case. 

It may be more than this without disadvantage other than 
the iucreased total leogth of the pair of expansion plates, but 
any reduction of this distance will cause each plate alternately 
to trespass upon the other's slot, at the extremities of the 
travel. 

Strictly speaking, the actual minimum permissible distance 
apart of the plates is governed by the steamway considered 
necessary for the admission of steam to the cylinder, when 
cutting off at the earliest specified point of the stroke ; the 
space between the two plates forming in effect a slot through 
which the steam has to pass. 

However, this is a needless refinement, and unless the 
internal length of the steam chest demands the limiting of the 
total length of the expansion plates to the barest possible 
dimension, we may take it as a safe rule that the slots should 
be so far apart that there is no trespassing or obstruction. 

This completes ouc observations in connection witih the 
expansion plates, and we may now proceed to construct a 

t i^iagram 45. 



128 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

suitable mam eccentric — that is, one which will give the 
proper steam way to correspond with the slot opening for the 
latest specified point of cut-off. From the general centre O 
(Diagram 45.) lay off upon the base-line A B, to the 
right, O P equal to four units, the specified lap of the main 
valve, and with radius O P describe the lap (semi ) circle 
K S P. From the point P lay off, still to the right, P Q equal 
to the given lead (one unit), and at the poiut Q erect a per- 
pendicular to the baselioe — the lead-line. The centre of the 
main eccentric will be somewhere on this line. If we find 
the amount of slot-opening when the crank is in the position 
corresponding to, say, fths of the stroke, and cause the main 
valve to have an equal opening at the same point, we shall 
have the full advantage of the slot-opening with no longer 
travel of the main valve than is just necessary to effect this 
purpose. 

To find the s/oi(-opening (the expansion plates being set for 
the maximum (|ths) cut-offj. when the crank is upon the 
radius O 3, corresponding to f ths of the piston's stroke : — 
With centre O, and radius O Z\ eqaal to six units (the pre- 
scribed extreme minus lap giving the fths cut off), describe a 
short arc Z^ t. Perpendicular to crank position O 8 draw its 
director Z^ io y. The slot is open a distance t Z'^ when 
the crank occupies the position O 3. 

Transfer this distance t Ti^ to the exterior of the main lap 
circle R S P on the same crank radius O'^ (i.e., lay off, on O 3, 
S T equal to t Tf). Perpendicular to O 3 at T draw the 
director, and the point C, where it cuts the lead-line Q 0, is 
the centre of the main eccentric to give the required port- 
opening at fths of the stroke. 

Join C, which will be the throw, and A O C will be the 
angular position of the main eccentric with respect to the 
real position of the crank at A. With centre O and radius 
O C describe the main travel circle A D C E. Touching the 
lap circle R S P and at right angles to the raclius O C draw 
the chord D E of the travel circle. Join O E (which will be 
the position of the crank when admission takes place) and 
O D (which will be its position when the main valve closes 
the port). Now make C X equal and parallel to Ojv, and 
join O X. 

O X will be the throw and A O X the angle with respect to 
the real position of the crank at A of the required expansion 
eccentric to fulfil the given conditions. 

Instead of taking the total 'permissible screw Tnovement of 
the Meyer expansion plate as the basis of operations, as in 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 129 

Diagram 45, we will assume that with a given minimum l^p 
in the expansion plate of six units {i.e., an extreme miuus^ 
lap of that amount) we desire to have a range of cut-offs 
varying from ^th up to ^ths of the piston stroke ; and that 
there shall be a steamway (or clear opening through both 
port and slot) of five units, at fths of the stroke, when the- 
plates are adjusted for their maximum cut-off of ^ths. 

As before, the lap of the main valve is to be four units, the 
lead one unit, and the single slot is to be of six units widths 
We have to find the laps of the expansion plate for cuu-offs 
at, say, ^th, ^, fths, i, and -^ths of the piston stroke (atiil- 
therefrom the amount of screw movement), and the throws' 
and angles of both eccentrics. 

With these requirements before us, we proceed to construct 
Diagram 45a, as follows : — 




Diagram 45a. 

In any straight line A B, let O be the centre of the crank- 
shaft, and O B the assumed starting point of the crank (its 
real direction of rotation being noted as before by the 
arrow). Let O 1 and O 4^ represent respectively the posi- 
tions of the crank at Jth and at ^ths of the piston's stroke, 
the earliest and latest desired points of cut-off; and let O 3 
represent the given position of the crank where the steam- 
way is to be the given amount of five units. 



130 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

With centre O and radius O Z^i, equal to six units, the 
given extreme minus lap, describe the short arc Z4| t^ and 
perpendicular to 0*1 draw the director touching the short arc 
and produce it to y, or farther. 

Upon O 3 lay off towards the centre O from the point t, the 
intersection of the arc, the distance t 7?, equal to five units, 
the desired slot-opening at phs. Perpendicular to O 3 draw 
the director at Z^ and produce it to y, the point of intersec- 
tion with the director Z^i, Join O y, which would be the 
throw (and k O y would be the angle with respect to the 
real position of the crank at A) of an eccentric which would 
fulfil the prescribed conditions if the main valve were fixed 
at the centre of its travel, forming a stationary face for the 
expansion plate to work upon. Now produce O 1, the crank 
position at the earliest desired point of cut-off (here ^th of the 
stroke), below the axis O, and draw perpendicular to it a 
director to y, the centre of the imaginary eccentric. The 
distance O Z^ from the centre to the point where this director 
cuts O 1 produced is the amount of ylus lap required to effect 
the Jth cut-off. 

From this point onwards the construction of Diagram 45a 
is identical with that of Diagram 45 (the two diagrams being 
lettered to correspond) ; and the matter of Case XLY., ante, 
from the words " Now make O x equal and opposite to O y'' 
onwards, is applicable throughout, substituting, of course, the 
laps and crank positions of Diagram 45a as they occur. 

Case XLVI. — Conditions as in Case XLV. To construct 
THE Zeuner Diagram, and from it the valve and expansion 
plates ; and to delineate the curves of steamway openings^ 

In Diagram 46, the throw and angle A O C of the main 
eccentric, and the throw and angle A O 3/ of the suppositional 
eccentric for moving the expansion plate upon the back of a 
fixed main valve, are those found by Case XLV. (Diagram 
45). The lap of the main valve is four units, the lead is one 
unit, the specified points of cut-off are ^th, ^, f ths, |-, and fths 
of the piston's stroke, reckoned from the assumed starting 
point B. The width of the single slots in the main valve is 
six units, the same as the steam port, and finally the amount 
of screw movement in each expansion plate, or the variation 
between the extremes of plus and minus lap, is 10 units — 
viz., from a plus lap of four units from the |th cut-off to a 
minus lap of six units for the fths cut-off. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 131 

Having Diagram 45 before us, we proceed to construct 
Diagram 46, as follows : — 




132 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

With centre O and radius O C describe the main travel 
circle A D C E, and upon O C as a diameter describe the 
main steam circle O R C P ; and a^ain, from centre O, with 
radius O P, describe the main lap circle L R P. Through the 
points of intersection P and R of these two circles draw O E 
and O D respectively. O E indicates the position of the 
crank when the main valve opens the port in the cylinder 
face, and O D its position at the instant of port closure (Case 
XVL). 

Upon O 5' as a diameter, describe the expansion circle O A 
y v., and draw in the radii (of the travel circle) O 5, O 4, 
O 3, O 2, and O 1, representing respectively the positions of 
the crank when the piston is at |ths, ^, fths, ^, and ith of its 
stroke reckoned from B. The points of intersection of these 
radii with the expansion circle O Ky N. are marked by the 
corresponding Roman numerals V., IV., III., II., and I. ; and it 
will be observed, in the case of the two last-named radii, that in 
order to intersect the circle they have to be produced below 
the centre O. The distances O V., O IV., and O III. indicate, 
being above the centre O, the respective minus laps of the 
expansion plate for cut-offs at the corresponding crank posi**^ 
tions ; while the distances O II. and O I., below the centre, 
denote the^j/ws laps necessary for the earlier points of cut-off 
corresponding to crank positions O 2 and O 1 respectively. 

If, now, we draw at any convenient distance below the 
base line A B a horizontal line g g to represent the upper 
surface of the main valve, and let fall the perpendicular O z 
upon it, the point z in the line g g will give us the cut-off edge 
of the slot. Arcs (shown dotted in the diagram) from the 
intersections O V., O IV., O III., O II., and O I., andperpendi- 
culars let fall from their contact with the base line A B, 
will give the successive laps for the corresponding points of 
cut-off — minus to the left of z and plus to the right of it. 

The determination of the length A of the plate is a simple 
matter. The left-hand expansion plate is shown in Diagram 
46, in its central position, and set for the earliest (Jth) cut-off. 
We have z a, the plus lap of four units, and we can lay down 
z c, the width of slot, which we know to be six units. What 
we have now to find is the length c & or the following lap of 
the plate necessary to avoid any reopening of the slot until 
after the ;port has been closed by the operation of the main 
valve. 

Let O D represent the position of the crank when this 
occurs. If now the length c & of the expansion plate be 
made equal to the distance O N^ [W being the point of inter- 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 133 

section of the crank radius O D with the expansion circle), 
the slot would be reopened simultaneously with the closing 
of the port. As a margin of safety of, say, ^in. is desirable, 
we may make c 6 = O L^ ; " and O F a crank radius drawn 
through the point Li will represent the position of the crank 
when the slot harmlessly reopens, the port being then over- 
closed to the extent of ^in. The total length A of the plate 
may thus be stated as az + zc + cb, ov extreme plus lap, 
plus slot width, plus the distance O L\ taken from the 
diagram. And the length B (or the distance between the 
inner edges of the slots) necessary to secure an unobstructed 
slot, may be stated as O jy + a. ^, or half the actual movement 
of the piate upon the back of the \a\Yeplus the extreme plus 
lap of the plate. 

All this is merely a confirmation of the results derived 
from the construction of Diagram 45. The real utility of 
Diagram 46 consists in the continuous representation, by 
means of the curves of steamway, of the combined movement 
of valve and plate, by which the opening and closing of ports 
and slots can be traced from start to finish of any admission 
period. We will trace that of the fths, or maximum, admis- 
sion period first. 

Premising that, as in previous diagrams, the horizontal 
shading of the crescent-shaped figure, bounded by the curves 
R C P and E, P, denotes by its radial breadth the amount the 
jyort is open at any crank position between O E and O D, the 
positions of opening and closure respectively, we proceed 
thus : — 

From the general centre O, with radius of six units (equal 
to the given slot width) describe the slot circle N v M to. The 
expansion plates, as we now suppose them to be adjusted, 
form for the present a plate having a minus lap of six units. 
As in the preceding Diagram 37,1 the crescent-shaped figure, 
included within the slot circle, but exterior to the expansion 
circle O Ay V., and shaded vertically, represents alot-opeu- 
ing, just as the before-mentioned horizontally-shaded crescent 
shows the port-opening. 

As will be seen by the shading, the slot is open through 
nearly two-thirds of the revolution, beginning when the crank 
(of course, proceeding in the opposite direction to the arrow) 
is upon the line O w, while through the half revolution 
extending from O Z^ to O Z, the slot is open its full width of 
six units (Zi Z being a line drawn at right angles to O y). 

■■ O L' being O N' plus Ihi. 
t Page 90. 



134 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

Slot-closure begins when the crank is on O Z, and ends, of 
course, vs^ith the arrival of the crank upon O 5, the specified 
point of cut-off, the curve denoting the closure being the arc 
of the expansion circle lying between these two crank posi- 
tions, or O V. 

Now we have very clearly, by means of the horizontally 
and vertically shaded crescents of Diagram 46, the respective 
openings of port and slot, shown separately, for any position 
of the crank within their range. In order to combine these 
two into one figure which shall show the actual clear steam- 
way during the whole period of admission, we have only to 
transfer this arc O V. to the exterior of the main lap circle 
L R P, so that it forms a bounding line of the steamway 
figure. 

This transference is effected precisely as in Diagram 37, by 
drawing in a few temporary crank-radii, measuring upon 
each the length intercepted between the lap circle (which in 
this case coincides with the slot circle J^ v M. iv) and the 
expansion circle O ky V.— in other words, the radial breadth 
of the vertical shading — and laying off this distance upon 
each radius exterior to the main lap circle L R P.t Through 
the points thus found the curve V^ Z is drawn. 

An examination of this steamway figure will show that up 
to the crank position O 3 the maia valve controls the steam 
passage ; at O 3 port and slot are each open an equal amount 
(the main eccentric was constructed to do this, in Diagram 
45), and after O 3 the expansion plate takes charge up to the 
point of cut-off. 

Now, if we screw the expansion plates until the minus lap 
is reduced to a little less thva four units, they will be in 
position for the ^ cut-off. The black curves, showing the 
steamway for each cut-off point, are drawn in on Diagram 46, 
but for the sake of clearness the method of finding the curves 
for the ^, fths, ^, and ^th cut-offs has been shown separately. 

In the small Diagram 46a the | cut-off curve is shown by 
itself. The main lap circle L R P, the main steam circle 
O R C P, and the expansion circle O lo y \J are those of 
Diagram 46, but only the lap circle IV. (shown dotted) is 
reproduced for the ^ cut-off. 

Upon the four temporary radii O a, O 6, O c, and O d, the 
respective lengths intercepted between the dotted lap circle 
and the expansion circle O w y U — viz., A A, B B, C C, and 
D D — are transferred io a a,h h, c c, and cl d, exterior to the 
main lap circle ; and through the points S a, b, c, d, the 

t See Case XXXVII. for fuller information, if required. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 135 




SCALI C*- (/."V/TS 



Diagram 46a— Steamway Curve for cut-off at § Stroke. 

curve is drawn. It will be noticed that the radius O d does 
not cut the expansion circle unless produced below the centre 
to D, but it is no exception to the rule of measuring from the 
lap circle to the point of intersection of the radius and the 
expansion circle. 




SCALE OF UNITS. 

Diagram 46b— Steamwat/Curve for cut-off at | Stroke. 



136 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 



Diagrams 46b, 46c, and 46d are constructed (for the fths, 
i, and the Jth curves respectively) in a similar manner, save 
that m the two latter, the lap being plus, the measurements 
are taken radially outwards from the lap circle, instead of 
inwards. 




Diagram 46c— Steamway Curve for cut-off at \ Stroke. 




scALe or UNITS 
Diagram 46d— Steamway Curve for cut-off at \ Stroke. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 137 

All this, which takes a long time to describe, is very readily 
done in actual practice, and it should be remembered that a 
complete diagram, such as No. 46, with all the movements 
of valve and expansion plates for each separate cut-off point 
permanently laid down upon it affords a more convenient 
and accurate means of trying over the valve gear than even 
a full-sized model would furnish, and at a fraction of the 
cost. 

Case XL VII. — Conditions as in Case XLV., but with 

DOUBLE or multiple -slotted VALVE. To CONSTRUCT THE 

VALVES AND ECCENTRICS FOR A GIVEN CYLINDER FACE TO 
FULFIL CONDITIONS AS SPECIFIED ; THE LAP OF MAIN VALVE, 
THE LEAD, THE EARLIEST AND LATEST POINTS OF CUT-OFF, AND 
THE WIDTH OF THE DOUBLE SLOTS, BEING GIVEN. 

The double-slotted main valve with its expansion plates for 
the Series D type is shown in Fig. 2. 

In Diagram 47 the throw and angle of the main eccentric, 
and the earliest and latest desired points of cut-off, correspond 
with those given for Case XLV. (Diagram 45), but instead of 
a single slot of 6 units in width, the valve has two slots of 3 
units each, and the expansion plate has a slot cut through it 
so as to give two cut-off edges. The amount of screw move- 
ment necessary to fulfil the given conditions is thereby 
reduced from 10 to 5 units for each plate, or a variation from 
2, X>ius lap of 2 units ( a , Fig. 2) to a minus lap of 3 units ; the 
former giving the minimum cut-off of ^th of the stroke, and 
the latter the maximum of fths of the stroke. 

We may now proceed to construct Diagram 47 (following 
the lines of Diagram 45 as nearly as possible), as follows : — 

In the straight line A B, let O be the centre of the crank- 
shaft, and O B the assumed starting point of the crank, its 
real direction of rotation being denoted, as before, by the 
arrow. Let O 1 and O 5 represent respectively the positions 
of the crank at ^th and at fths of the piston's stroke, the 
earliest and latest desired points of cut-off. Lay off, upon 
O 5, O Z^, equal to 3 units, the prescribed extreme minus lap 
of the expansion plate, and, perpendicular to O 5, draw the 
director at Z^ producing it to y, or farther. 

Upon O 1 produced below the axis O, lay off similarly, but 
downwards, the distance O Z^ equal to 2 units, the prescribed 
extreme plus lap, and, perpendicular to O 1, draw the director 
at Z^, and produce it to y, the point of intersection with the 
director Z^ Join O y, which would be the throw (and AO y 
would be the angle with respect to the real position of the 



138 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 




\ SC/iL£ o£ VfilTS 

Diagram 47. 
crank at A) of an eccentric which would fulfil the prescribed 
conditions with the given laps, if the main valve were fixed at 
the centre of its travel, forming a stationary face for the 
expansion plates to work upon. 

Here O j^ is of course exactly half the length of O 3/ in 
Diagram 45, and were it not for the obstruction of the second 
or outer slot due to the passing over it of the inner bar A of 
the expansion plate Fig. 2, we should, with our two 3-unit 
slots, get precisely the same total slot-opening as in Diagram 
45 with the single 6-unit slot. 




>8 — a*_3e- 

SC^LE OF U/V1T5 



Fig. 2, Series D— Valve with Double Slots and Series D 
JixPANSioN Plate, 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 139 

We have already discussed this matter pretty fully (Diagram 
32b ante), and have ascertained that by cutting away the bach 
on folloivivg edge of the slot in the expansion plate we can 
cause the outer slot to begin its opening as early in the stroke 
as may be necessary. In the present case we will assume 
that when the plates are screwed to their extreme distance 
apart, to give the f ths or maximum cut-off, it is desired that 
the outer slot shall begin to open when the crank is on the 
dead centre O B — i.e., at zero of the stroke. 

It may be noted that it is in general only with reference to 
the maximum point of cut-off that the delayed opening of the 
second slot need be considered at all, because the obstruction 
rapidly lessens as the admission period is reduced. 

To resume the construction of Diagram 47. Make O x 
equal and opposite to O y, and upon y x, 3,^ o, diameter, 
describe a semi-circle. At right angles to y x draw a radius 
of this semi-circle O z, and through the point z draw the 
horizontal line g g, which will serve to represent the upper 
surface of the main valve (g g, Fig. 2) upon the actual scale of 
the diagram. 

If now we take the point z, in the straight line g g, to 
represent the edge z of the inner left-hand slot in Fig. 2, we can 
construct the left-hand expansion plate (of which, of course, 
the right-hand plate is a reversed facsimile) directly from 
the diagram. 

An arc struck from centre z in the semi-circle to touch the 
crank radius O 1 will cut the line g g at a, and the distance 
z a will be the lap required to effect the ^th cut-off, which 
was to be 2 units. Another arc struck from centre z to touch 
the crank position O 5 will in like manner cut the line g g at 
the point c. The distance z c will indicate the lap required 
for the fths, or latest desired cut-off, which was to be 3 units. 
And, similarly, for intermediate positions of the crank, as O 4, 
O 3, and O 2, arcs struck from z, touching those crank radii, 
will give, by their intersections with the line g g, the required 
laps for the respective points of cut-off— viz., ^, fths, and ^ 
of the piston stroke ; while a lap of O {i.e., when the cut-off 
edge a of the expansion plate has, by a screw movement of 2 
units, been brought into coincidence with the inner edge z of 
the valve slot) corresponds with a cut-off when the crank 
occupies the position O z. Laps to the right of the point z are, 
of course, positive or plus laps, and those to the left of z, 
negative or minus. 

We have now upon the horizontal line g g the laps clearly 
shown for each eighth of the piston's stroke between the 
specified limits of ^th and fths, and these are all comprised 



140 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

within the given range of screw movement, which was to be 
half that specified for Diagram 45, or 5 units — viz., 2 units of 
plus, and 3 units of minus lap. We have now to find the 
dimensions of the various parts of the expansion plate, for it 
will be noted that hitherto we have only determined jposi^ions, 
or screw movements. 

The length A of the bars in the expansion plate is deter- 
mined precisely as in Case XLV., and must be fixed with 
reference to the position of the plates when they are placed as 
shown in Fig. 2 — i.e., at their maximum of plus lap. 

Measuring from the leading edge a of the plate, we know 
<x ^ to be 2 units, and we know a- c to be 5 units (as the width 
of slot ^^ c is three units by the given conditions). We have 
therefore to find the distance c b, or what may be called the 
trailing lap of the plate. 

Let O D represent the position of the crank at the instant 
the main valve closes the port in the cylinder face. (We shall 
fix this crank position O D presently, but for the moment we 
may assume it to be given, or determined). From the point z 
(in the semi-circle v z x) describe an arc N^ N touching this 
crank radius O D. Now if the distance c b were made equal 
to the radius of this arc, we should have a reopening by the 
expansion plate at the precise instant that the main valve had 
closed the port. Taking as before a margin of safety of ^in., 
if we make the length or distance c 6 in the expansion plate 
Jin. in excess of the radius of arc N^ N, the desired margin of 
safety will be secured ; and the crank radius O F, drawn to 
touch arc Li L (which is of Jin. greater radius than arc N^ N), 
will indicate the position of the crank when the slot harm- 
lessly re-opens. 

We have, then, as the length A of each of the bars of the 
expansion plate, az + zc + cb, where a z is the extreme 
plus lap, z c the width of slot in the valve, and c b the radius 
of an arc struck from centre z to touch a crank position O F 
later in the stroke than the position O D, at which the main 
valve closes. 

We have now to determine the width of the slot 6 ^ in the 
plate, separating the two bars marked A (Fig. 2). This 
depends entirely upon the point in the stroke at which we 
wish the second or outer slot to begin its opening. As already 
mentioned in the earlier part of the present case, we assume 
that with the expansion plates screwed to their extreme dis- 
tance apart, it is required that the outer slot shall begin to 
open when the crank is on the dead centre O B.^' 

* The first or inner slot opens very much earlier in the revolution, beginning 
when the crank occupies the positioa O w (see Diagram 47a). 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 141 

To secure this result we have only to make the width of 
slot in the expansion plate equal to the radius of an arc, 
struck from centre z, touching the crank position at which 
the slot is to begin its openiDg. In the present case we make 
the slot width h d equal to the radius of the arc S U, which 
touches crank position O B. The slot will continue to open 
until the crank reaches the position O J^ and will remam wide 
open while the crank passes through the arc of revolution 
Ji z after which, both slots gradually close, cut-off occurring of 
course with the crank at O 5. This period J^ z (during which 
the outer slot remains wide open) is defined by the radius of 
arc yi V, struck from centre z, which corresponds with the 
distance 6 n in the expansion plate slot, and is 3 units (or the 
width of the valve-^lot) less than the radius of arc S U. 

Supposing for a moment that the width of slot in the plate 
were made 3 units, the same as the valve-slots, then the outer 
slot would only begin to open when the crank reached the 
position O J ; O J being the radius of a crank circle drawn to 
touch the arc T^ T, struck from centre z with radius 3 units. 
In this case there would be no period like Ji z of full opening, 
but the slot would be opening during the period J z, and 
closing during the period z 5, in the revolution of the crank. 

This opening and closing of the second, or outer, slot may 
perhaps be rendered clearer by a consideration of the small 




SC/iL£ OF UNITS 



Diagram 47a, 



diagram 47a, in which the horizontally-shaded portion re- 
presents by its radial breadth the amount this outer slot is 
open at any crank position in the revolution between O B, 



142 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

where opening begins, and O 5, vp^here closure occurs, the 
vertical shading representing the opening due to the first or 
inner slot. 

Diagram 47a is derived from Diagram 47, and the lettering 
has the same significance, the only differences being that the 
arcs yi V, T^ T, and S U are struck from centre instead of 
from z, and that upon O x and O j' as diameters the Zeuner 
circles are described. As the crank revolves we see the 
breadth of horizontal shading, which is nothing when the 
crank is on the dead centre O B, increase gradually up to 3 
units at crank position O Ji, remain at 3 units breadth until 
the crank reaches the position O z, and then (the vertical 
shading from O ^' onward being common to both slots) gradually 
narrow to nothing again as the crank approaches the position 
of cut-off O 5. 

After this digression to demonstrate by independent 
evidence the correctness of our method, we may continue the 
construction of the valve and expansion plates. 

The total length of each plate is evidently made up of two 
bars (A, A, Fig. 2) separated by a gap or slot whose width is 
equal to the radius of arc S U (Diagram 47 or 47a). The 
length of bar A we have already determined. We have now 
to fix the minimum distance B separating the two inside 
edges z, z of the inner slots in the valve. If this distance B 
be insufficient, the left-hand slot may be partially covered by 
the right-hand plate when the plates are at the extremity of 
their travel to the left, or vice versa. 

The minimum distance B necessary to secure an un- 
obstructed slot is one-half the actual movement of the plate 
upon the back of the valve, plus the extreme plus lap of the 
plate, or in this case O y\ + 2 units. It may be more than 
this without disadvantage, but any reduction of this distance 
will cause each plate alternately to trespass upon the other's 
slot at the extremities of the travel. 

One more dimension of the main valve remains to be 
determined, the distance C ( Fig. 2, and Diagram 47). C is 
the distance separating the two slots, the inner and the outer, 
at either end of the valve, and is equal to the bar length A, 
plus the distance b n, or the radius of arc V^ V. This 
completes the dimensions of the valve and plates, and we may 
now proceed to construct a suitable main eccentric — that is, 
one which will give the proper steam way to correspond with 
the slot-opening for the latest specified point of cut off. 

t Diagram 47. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 143 

From the general centre O, lay off upon the base line A B 
to the right, O P equal to 4 units, the specified lap of the 
main valve, and vpith radius O P describe the lap (semi) circle 
E G P. From the point P lay off, still to the right, P Q equal 
to the given lead (1 unit), and at the point Q erect the lead 
line perpendicular to the base line. The centre of the main 
eccentric will be somewhere on this line. If we find the 
amount of slot-opening when the crank is in the position 
corresponding to, say, |ths of the stroke, and cause the main 
valve to have an opening equal to the combined opening of 
the two slots at this point, we shall have the full advantage 
of the slot-opening with no longer travel of the main valve 
than is just necessary to effect this purpose. 

To find the s/o^opening (the expansion plates being set for 
maximum — |ths — cut-off), when the crank is upon the radius 
O 3, corresponding to fths of the piston's stroke : — With centre 
O, and radius O Z^, equal to 3 units (the prescribed extreme 
minus lap giving the §ths cut-off), describe a short arc Z^ t. 
Perpendicular to crank position O 3 draw its director Z^ to y. 
The slot is open a distance t 7i^ when the crank occupies the 
position O 3. 

Transfer double"^' this distance, t Z^, to the exterior of the 
main lap circle R G P on the same crank radius O 3 — i.e., lay 
off on O 3, G H equal to tivice t 7P. Perpendicular to 3 at 
H draw the director, and the point C, where it cuts the lead 
line Q C, is the centre of the main eccentric to give the 
required port-opening at fths of the stroke. 

Join O C ; C will be the throw, and A O C the angle with 
respect to the real position of the crank at A, of the required 
main eccentric to fulfil the given conditions. With centre O 
and radius O C describe the main travel circle A D C E. 
Touching the lap circle R G P and at right angles to the radius 
O C, draw the chord D E of the travel circle. Join O E 
(which will be the position of the crank when admission takes 
place) and O D (which will be its position when the main 
valve closes the port). 

Now make C X equal and parallel to O j, and join O X. 
O X will bs the throw, and A O X the angle with respect to 
the real position of the crank at A, of the required expansion 
eccentric to fulfil the given conditions with the double-slotted 
valve and the expansion plates of Diagram 47. 

Fig. 1 illustrates the outside or Series E type of Meyer 
expansion plates, shown in the position in which the two 
halves are supposed to be when at their greatest distance 

* For a two-slotted valve. 



144 THE PROPORTIONS AND MOVEMENT OF SLIDE TALVES. 




• -^I D <2 /a 2f JO aa 42 — 49 « M 

SCALBi OF UNJTS. 

Fig. 1, Series E— Valve avith Double Slots and Series E 
Expansion Plate, 

apart, having then Si, plus or positive lap equal to the distance 
A . The main valve is of the same character as that illus- 
trated in Fig. 1, Section VI. t 

The plates, when extended as shown, effect their earliest 
cut-off ; and when screwed close together admit the steam up 
to the latest desired point of cut off, the plus lap changing to 
minus as the plates approach each other. 

As noticed in the preceding section, in cases where the 
variable expansion gear is automatic {i.e., controlled by the 
action of a governor), it is commonly advisable to arrange for 
a zero cut-off, otherwise the range may vary from, say, ^th 
of the stroke onwards ; it being always advisable to avoid an 
excessive amount of in-and-out screwing when changing from 
one degree of expansion to another. 

Being an outside expansion plate, the expansion eccentric 
is set in advance of the crank, as in the Series A and Series C 
types. 

The conditions specified in the present series are through- 
out precisely those of Series D, so as to afford an opportunity 
of comparing the throws and angles of the respective expan- 
sion eccentrics in the corresponding cases of the alternative 
systems. 

t See page 87. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 145. 

Case XLVIII. — Variable expansion by altering the laf 

OF THE EXPANSION PLATE, THE THROW AND ANGLE OF THE 
EXPANSION ECCENTRIC BEING FIXED. To CONSTRUCT THE 
VALVES AND ECCENTRICS FOR A GIVEN CYLINDER FACE TO FULFIL 
CONDITIONS AS SPECIFIED, THE LAP OF MAIN VALVE, THE LEAD, 
THE EARLIEST AND LATEST POINTS OF CUT-OFF, AND THE WIDTH 
OF THE SINGLE SLOT, BEING GIVEN. 

As before, we will take in the first instance, as the condition" 
to be fulfilled, an assumed restriction as to the amount of 
screw movement permissible ; say 10 units as a maximum fof 
each half of the expansion plate. 

We may apportion this as follows — viz., a variation from a 
2Jhcs lap of 4 units (a. Fig. 1) to a minus lap of 6 units, the 
latter giving, of course, the latest desired cut-off, which we 
will assume to be at fths of the piston's stroke. The earliest 
cut-off is to be at ^th, and the width of the single slots in the 
valve is taken as 6 units. 

We may now proceed to construct Diagram 48 as follows: — 




SCALE OP nNJTS. 



Diagram 48. 
In the straight Hne A B let O be the centre of the crank- 
shaft, and O B the assumed starting point of the crank (its 
real direction of rotation beiog denoted, as before, by the 
arrow). Let O 1 and O 5 represent respectively the positions 



146 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES 

of the crank at Jth aud at fths of the piston's stroke, the 
earliest and latest desired points of cut-off. Lay off upon O 1, 
O Zi equal to 4 units, the prescribed extreme plus lap, and 
perpendicular to O 1 draw the director at Z^, producing it to 
X, or farther. 

Upon O 5 produced below the axis O, lay off similarly, but 
downwards, the distance O Z^ equal to six units, the prescribed 
minimum lap {i.e., the extreme minus lap), and perpendicular 
to O 5, draw the director at Z^, and produce it to x, the point 
of intersection with the director Z^. Join O x, which would 
be the throw (and kO x would be the angle, with respect to 
the real position of the crank at A) of an eccentric which would 
fulfil the prescribed conditions, with the given laps, if the main 
valve were fixed at the centre of its travel, forming a 
stationary face for the expansion plates to work upon. As 
before, we may conveniently proceed upon this assumption 
for the present, leaving the main eccentric and the motion of 
the main valve to be given, or determined, later on in the 
case. 

Now make Oy equal and opposite to O x, and upon y x, as 
a diameter, describe a semi-circle. At right angles io y x draw 
a radius of this semi-circle O z. 

O z represents the position of the crank (assumed to start 
from B) where the steam would be cut off with no lap, either 
plus or minus, of the expansion plate — i.e., when the cut-off 
edges a, a, Fig. 1, have, by a screw movement of 4 units, been 
brought into coincidence with the outer edges z, z of the valve 
slots, a position identical with that of the fixed expansion 
plate in Series A. It is, of course, understood that in the 
present series, lap, whether plus or minus, is measured from 
these outer or cut-off edges z, -z of the valve slots, when the 
valve and the plates are in mid-travel, and their centres 
co-incident. 

Through the point z in the semi-circle just found draw the 
horizontal straight line g g, which will serve to represent the 
upper surface of the main valve {g g. Fig. 1) upon the actual 
scale of the diagram. 

If now we take the point z in the straight line g g, in Dia- 
gram 48 to represent the outer edge z of the right-hand slot 
in Fig. 1, we can construct the right-hand expansion plate (of 
which the left-hand plate is a veyeicsed facsimile) directly from 
the diagram. 

An arc struck from centre z to touch the crank radius O 1 
will cut the line g g a,i a, and the distance z a will be the lap 
required to effect the ^th cut-off, which was to be 4 units. 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 147 

Another arc struck from centre z to touch the crank radius 
O 5 will cut the line g g at the point c. The distance z c will 
indicate the minus lap required for the f ths or latest desired 
cut-off, which was to be 6 units, and similarly for intermediate 
positions of the crank, as O 4, O 3, and O 2^ arcs struck from z, 
touching those crank radii, will give, at their intersections 
with the line g g, the required laps for the respective points 
of cut-off — viz., i, fths, and a ^th of the piston's stroke ; while, 
as just now noticed, a lap of O, or zero, corresponds with a 
cut-off when the crank occupies the position O z. Laps to the 
right of the point z are of course positive or plus laps, and 
those to the left of z negative or minus. 

We have now upon the horizontal line g g the laps clearly 
shown for each eighth of the piston's stroke between the 
specified limits of ^th and fths, and these are all comprised 
within the given range of screw movement, which was to be 
10 units — viz., 4 units of plus and 6 units of minus lap. We 
have now to find two important dimensions, for it will be noted 
that hitherto we have only determined positions, or screw 
movements. We want to get the dimensions denoted by A 
and by B, Fig. 1, both of which require careful consideration. 

If the length of the expansion plate A be insufficient, the 
slot will be reopened or uncovered by the trailing edge b of the 
plate before the port in the cylinder face has been closed by 
the main valve. This would cause a readmission of steam 
during the period of expansion. This length A must, of course, 
be determined with reference to the position of the plates when 
they are placed as shown in Fig. 1—i.e., at their maximum of 
plus lap. 

Measuring from the leading edge a of the plate, we know a 
^ to be 4 units, and we know a c to be 10 units (as the width 
of slot z c is 6 units by the given conditions). We have there- 
fore to find the distance c b, or what may be called the trailing 
lap of the plate. 

Let O D represent the position of the crank at the instant 
the main valve closes the port in the cylinder face.''' From 
the point z describe an arc N^ N touching the crank radius O 
D. Now, if the distance c b were made equal to the radius of 
this arc, we should have a reopening by the expansion plate 
at the precise instant tbat the main valve had closed the port. 

A margin of safety is desirable, and for this purpose ^in., 
whether the valve be a large or a small one, is not too much. 
If, therefore, the length c 6 of the expansion plate be made |iQ. 

"* We have not fixed this or any other operation of the main valve yet, but fo 
the moment we may consider it to be given or determined. 



148 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES, 

in excess of the length of radius of arc N^ N, the desired 
margin of safety will be secured, and the crank radius O F, 
drawn to touch arc L^ L (which is of Jin. greater radius than 
arc N^ N), will indicate the position of the crank when the slot 
harmlessly reopens. 

We have, then, as the total length A of the expansion plate 
a z + z c -\- c b, where a z is the extreme plus lap, z c the 
width of slot in the valve, and c h the radius of an arc struck 
from centre z to touch a crank position O F later in the stroke 
than the position O D, at which the main valve closes. 

We have now to determine the minimum distance B sepa- 
rating the two inner edges c, c of the slots in the main valve 
CFig. 1). It is obvious that this is simply twice the bar length 
A of the expansion plate, the only condition being that the 
pair of plates, when screwed close together, shall have a minus 
lap of not less than 6 units, as prescribed. With the plates 
in this position, they are in effect identical with the fixed 
expansion plate, Series C, illustrated on page 87. 

This distance B between the slots may, of course, be 
increased without disadvantage, provided, there be room in 
the steam chest for the plates to travel. 

This completes our observations in connection with the 
expansion plates, and we may now proceed to construct a 
suitable main eccentric — that is, one which will give the 
proper steamway to correspond with the slot-opening for the 
latest specified point of cut off. 

From the general centre O, lay off, upon the base line A B, 
to the light, O P, equal to 4 unitS; the specified lap of the main 
valve, and with radius O P describe the lap (semi-) circle 
R S P. From the point P lay off, still to the right, P Q equal 
to the given lead (1 unit), and at the point Q erect a perpen- 
dicular to the base line — the lead line. The centre of the 
main eccentric will be somewhere on this line, If we find the 
amount of slot-opening when the crank is in a position corres- 
ponding to, say, f ths of the stroke, and cause the main valve 
to have an equal opening at the same point, we shall have the 
full advantage of the slot-opening with no longer travel of the 
main valve than is just necessary to effect this purpose. 

To find the sZotopening (the expansion plates being set for 
the maximum (f ths) cut-off) when the crank is upon the radius 
O 3, correspondmg to gths of the piston's stroke : — 

With centre O and radius O Z^, equal to 6 units (the pre- 
scribed extreme mmus lap giving the fths cut-off), describe a 
short arc Z^ t. Perpendicular to crank position O 3 produced 
below the axis O, draw the director Z^ to the point x. The 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 149 

slot is open a distance t Z'^ when the crank occupies the 
position O 3. Transfer this distance t Ta^ to the exterior of 
the main lap-circle R S P, on the crank radius O 3— ^.e., lay 
off upon O 3, S T equal to t Z=^. Perpendicular to O 3, at T 
draw the director, and the point C, where it cuts the lead line 
Q C, is the centre of the main eccentric to give the required 
port-opening at f ths of the stroke. 

Join O C, which will be the throw, and A O C will be the 
angular position of the main eccentric with respect to the real 
position of the crank at A. With centre O, and radius O C, 
describe the main travel circle A D C E. Touching the lap 
circle R S P, and at right angles to the radius O C, draw the 
chord D E of the travel circle. Join O E (which will be the 
position of the crank when admission takes place), and O D 
(which will be its position when the main valve closes the 
port). Now make C X equal and parallel to O x, and join 
O X. 

O X will be the throw, and A O X the angle with respect to 
the real position of the crank at A, of the required expansion 
eccentric to fulfil the given conditions. 

Instead of taking the total 'permissible screw movemevX of 
the Meyer expansion plate as the basis of operations, as in 
Diagram 48, we will assume that with a given mioimum lap 
in the expansion plate of 6 units {i.e., an extreme minus lap 
of that amount), we desire to have a range of cut-offs varying 
from ith up to -^ths of the piston's stroke ; and that there 
shall be a steamway (or clear opening through both port and 
slot) of 5 units at fths of the stroke, when the plates are 
adjusted for their maximum cut-off of ^ths. 

As before, the lap of the main valve is to be 4 units, the 
lead 1 unit, and the single slot is to be of 6 units width. We 
have to find the laps of the expansion plate for cut-offs at, 
say, ^th, \, fths, \, and ^ths of the piston's stroke (and there- 
from the amount of screw movement), and the throws and 
angles of both eccentrics. 

With these requirements before us, we proceed to construct 
Diagram 48a as follows : — 

In any straight line A B, let O be the centre of the crank- 
shaft, and O B the assumed starting point of the crank (its 
real direction of rotation being denoted, as before, by the 
arrow). Let O 1 and O 4^ represent respectively the positions 
of the crank at Jth and at ^ths of the piston's stroke, the 
earliest and latest desired points of cut-off ; and let O 3 
represent the given position of the crank where the steamway 
is to be the given amount of 5 units. 



150 THE PROPORTIONS AND MOVEMENT OF SLUE VALVES. 




SCALE QF UNITS. 



Diagram 48a. 



With centre O and radius Z^i, equal to 6 units, the given 
extreme minus lap, describe the short arc Z^i t, and, per- 
pendicular to O 4:^ produced, draw the director touching the 
short arc, and produce it to x or farther. 

Upon O 3, produced below the axis O, lay off towards O 
from the poiut t, the intersection of the arc, the distance t 71^^ 
equal to 5 units, the desired slot openiog at fths. Per- 
pendicular to O 3 produced draw the director at Z^, and 
produce it to x, the point of intersection with the director Z^. 
Join O X which would be the throw (and KO x would be the 
angle with respect to the real position of the crank at A) of 
an eccentric which would fulfil the prescribed conditions if 
the main valve were fixed at the centre of its travel, forming 
a stationary face for the expansion plate to work upon. 

Now, perpendicular to O 1, the crank position at the earliest 
desired point of cut-off (here Jth of the stroke), draw a 
director to x, the centre of the imaginary eccentric. The 
distance O Zi from the centre to the point where this director 
cuts O 1, is the amount of j^lus lap required to effect the Jth 
cut-off. 

From this point onwards the construction of Diagram 48a 
is identical with that of Diagram 48 (the two diagrams being 
lettered to correspond), and the matter of Case XLVIII. ante^ 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 151 

from the words, " now make O T equal and opposite to 0,r," 
onwards, is applicable throughout, substituting, of course, the 
laps and crank positions of Diagram 48a as they occur. 

Case XLIX — Conditions as in Case XLVIII. To construct 

THE ZeUNER diagram, AND FROM IT THE VALVE AND EXPANSION 
plates ; AND TO DELINEATE THE CURVES OF STEAMWAY OPENINGS. 

In Diagram 49 the throw and angle A O C of the main 
eccentric, and the throw and angle A O ;v of the suppositional 
eccentric for moving the expansion plate upon the back of a 
fixed main valve, are those found by Case XLVIII. (Diagram 
48). The lap of the main valve is four units, the lead is one 
unit, the specified points of cut-off are ^th, ^th, |ths, ^, and 
;|ths of the piston's stroke, reckoned from the assumed start- 
ing point B. The width of the single slots in the main valve 
is six units, the same as the steam port, and finally the 
amount of screw movement in each expansion plate, or the 
variation between the extremes of plus and minus lap, is 10 
units — viz., from a plus lap of four units for the ^th cut-off to 
a minus lap of six units for the §ths cut-off. 

Having Diagram 48 before us, we proceed to construct 
Diagram 49 as follows : — 




Diagram 49. 



152 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

With centre O and radius O C describe the main travel- 
circle A D C E, and upon O C as a diameter describe the 
main steam-circle O R C P, and again from centre O with 
radius O P describe the main lap-circle L R P. Through the 
points of intersection P and R of these two circles draw O E 
and O D respectively. O E indicates the position of the 
crank when the main valve opens the port in the cylinder 
face, and O D its position at the instant of port-closing (Case 
XVI.). 

Upon O jK as a diameter describe the expansion-circle O A 
y V and draw in the radii O 5, O 4, O 3, O 2, and O 1, repre- 
senting respectively the positions of the crank when the 
piston is at §ths, -k, fths, ^th, and Jth of its stroke, reckoned 
from B. The points of intersection of these radii with the 
expansion-circle O A y Y are marked by the corresponding 
Roman numerals V., IV., III., II., and I., and it will be 
observed, in the case of the two last-named radii, that in 
order to intersect the circle they have to be produced below 
the centre O. The distances O V., O IV., and O III. indicate, 
being above the centre O, the respective minus laps of the 
expansion plate for cut-offs at the corresponding crank posi- 
tions; while the distances O II. and O I., 6e/o?(; the centre, 
denote the plus laps necessary for the earlier points of cut-off 
correspondmg to crank positions O 2 and O 1 respectively. 

If now we draw at any convenient distance below the base 
line A B a horizontal line g g to represent the upper surface 
of the main valve, and let fall the perpendicular O z upon it, 
the point z in the line g g will give us the cut-off edge of the 
slot. Arcs (shown dotted in the diagram) from the inter- 
sections O v., O IV., O III., O II., and O I., and perpendiculars 
let fall from their contact with the base line A B, will give 
the successive laps for the corresponding points of cut-off — 
minus to the left of z and plus to the right of it. 

The determination of the length A of the plate is a simple 
matter. The right-hand expansion plate is shown in Diagram 
49, in its central position, and set for the earliest (^th) cut off. 
We have z a the plus lap of 4 units, and we can lay down z c 
the width of slot, which we know to be 6 units. What we 
have now to find is the length c b, or the following lap of the 
plate, necessary to avoid any reopening of the slot, until after 
the port has been closed by the operation of the main valve. 

Let O D represent the position of the crank when this 
occurs. If now the length c b oi the expansion plate be made 
equal to the distance O N^ (N^ being the point of intersection 
of the crank radius O D with the expansion circle), the slot 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 153 

"would be reopened simultaneously with the closing of the 
port. As a margin of safety of, say, Jin. is desirable, we may 
make c 6 = O L^ ;''' and O F, a crank radius drawn through 
the point L^, will represent the position of the crank when 
the slot harmlessly reopens, the port being then over-closed 
to the extent of Jin. The total length A of the plate may 
thus be stated as az + zc + cb, or extreme plus lap, pllis 
slot width, plus the distance O L^ taken from the diagram, and 
the length B (or the distance between the inner edges of the 
slots) may be stated as twice the length A of the expansion 
plate, as a minimum. It may be increased at convenience 
(for example, the slots in Fig. 1 might be straight 
instead of being curved inwards, as shown) without dis- 
advantage, provided there be room in the steam-chest for the 
plates to travel. 

From this point onwards the construction of Diagram 49 is 
identical with that of Diagram 46, and the matter and 
diagrams of Case XLVL, from the words " Premising that as 
in previous diagrams," may be followed exactly in completing 
the present case. 

Case L. — Conditions as in Case XLVIIL, but with double 

OR MULTIPLE SLOTTED VALVE. To CONSTRUCT THE VALVES AND 
ECCENTRICS FOR A GIVEN CYLINDER FACE TO FULFIL CONDITIONS 
AS SPECIFIED ; THE LAP OF MAIN VALVE, THE LEAD, THE EARLIEST 
AND LATEST POINTS OF CUT-OFF, AND THE WIDTH OF THE DOUBLE 
SLOTS BEING. 




acALZ OF UNne. 



Fig. 



Series E— Valve avith Double Slots and Series E 
Expansion Plate. 



O L^ being O N' -plus \va.. 



154 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

The double-slotted main valve, with its expansion plate for 
the Series E type, is shown in Fig. 2. 

In diagram 47 the throw and angle of the main eccentric, 
and the earliest and latest desired points of cut-off, correspond 
with those given for Case XLVIII. (Diagram 48), but instead 
of a single slot of 6 units in width, the valve has two slots of 
3 units each, and the expansion plate has a slot cut through 
it po as to give two cut-off edges. The amount of screw 
movement necessary to fulfil the given conditions is thereby 
reduced from 10 to 5 units for each plate, or a variation from 
a, plus lap of 2 units (a , Fig. 2) to a tninus lap of 3 units, the 
former giving the minimum cut-off of Jth of the stroke, and 
the latter the maximum of fths of the stroke. 

We may now proceed to construct Diagram 50 (following 
the lines of Diagram 48 as nearly as possible), as follows : — 

In the straight line A B let O be the centre of the crank- 
shaft, and O B the assumed starting point of the crank (its 
real direction of rotation being denoted, as before, by the 
arrow). 

Let O 1 and O 5 represent respectively the positions of the 
crauk at ^th and at f ths of the piston's stroke, the earliest and 
latest desired points of cut-off. Lay off upon O 1, O Z^ equal 
to 2 units, the prescribed extreme plus lap, and perpendicular 
to O 1 draw the director at Z^, producing it to x or farther. 

Upon O 5, produced below the axis O, lay off similarly, but 
downwards, the distance O Z^ equal to 3 units the prescribed 
minimum lap {i.e., the extreme minus lap), and perpendicular 
to O 5 draw the director at Z^ and produce it to x, the point 
of intersection with the director Z^. Join O x., which would 
be the throw (and K O x would be the angle with respect to 
real position of the crank at A) of an eccentric which would 
fulfil the prescribed conditions with the given laps if the 
main valve were fixed at the centre of its travel, forming a 
stationary face for the expansion plates to work upon. 

Here O x is, of course, exactly half the length of O at in 
Diagram 48, and were it not for the obstruction of the second 
or inner slot due to the passing over it of the bar A of the 
expansion plate (Fig 2), we should, with our two 3 -unit slots, 
get precisely the same total slot opening as in Diagram 48 
with the single 6-unit slot. 

We have already discussed this matter pretty fully (Dia- 
grams 24b and 38b, ante), and have ascertained that by 
cutting away the &<xcZf or following edge of the slot in the 
expansion plate we can cause the inner slot to begin its 
opening as early in the stroke as may be necessary. In the 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 155 

present case we will assume that when the plates are screwed 
to their minimum distance apart to give the fths or maximum 
cut-off, it is desired that the inner slot shall begin to open 
when the crank is on the dead centre O B — i.e., at zero of the 




156 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

stroke. [It may be noted that it is in general only with 
reference to the maximum point of cut-off that the delayed 
opening of the second slot need be considered at all, because 
the obstruction rapidly lessens as the admission period is 
reduced.] 

To resume the construction of Diagram 50 : Make O y equal 
and opposite to O x, and upon ^ ;r, as a diameter, describe a 
semi-circle. At right angles to y x draw a radius of this semi- 
circle o z, and through the point z draw the horizontal line 
g g, which will serve to represent the upper surface of the 
main valve {g g, Fig. 2) upon the actual scale of the diagram. 

If, now, we take the point z in the straight line g g to 
represent the edge z of the outer right-hand slot in Fig. 2, we 
can construct the right-hand expansion plate (of which, of 
course, the left-hand plate is a reversed facsimile) directly 
from the diagram. 

An arc struck from centre z in the semi-circle to touch the 
crank radius O 1, will cut the line ^ ^ at <x, and the distance 
z a will be the lap required to effect the gth cut-off, which 
was to be 2 unils. Another arc struck from centre z to touch 
the crank position O 5, will in like manner cut the line ^ ^ at 
the point c. The distance z c will indicate the lap required 
for the fths or latest desired cut off, which was to be 3 units. 
And, similarly, for intermediate positions of the crank, as 
O 4, O 3, and O 2, arcs struck from z touching those crank 
radii will give by their intersections with the line g g the 
required laps for the respective points of cut-off — viz., ^, fths, 
and :^th of the piston stroke ; while a lap of O (i.e., when the 
cut-off edge a of the expansion plate has, by a screw move- 
ment of 2 units, been brought into coincidence with the outer 
edge z of the valve slot) corresponds with a cut-off when the 
crank occupies the position O z. Laps to the right of the 
point z are, of course, positive or plus laps, and those to the 
left of z negative or minus. 

We have now upon the horizontal line g g the laps clearly 
shown for each eighth of the piston's stroke between the 
specified limits of ^th aud |ths, and these are all comprised 
within the given range of &crew movement, which was to be 
half that specified for Diagram 48, or 5 units — viz. ,2 units of 
plus, and 3 units of minus lap. We have now to find the 
dimensi07is of the various parts of the expansion plate, for it 
will be noted that hitherto we have only detevmined positions, 
or screw movements. 

The length A of the bars in the expansion plate is deter- 
mined precisely as in Case XL VIII., and must be fixed with 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 157 

reference to the position of the plates when they are placed 
as shown in Fig. 1—i.e., at their maximum of plus lap. 

Measuring from the leading edge a of the plate we know 
a z io he two units, and we know (X c to be 5 units (as the 
width of slot 2- c is 3 units by the given conditions). We have, 
therefore, to find the distance c b, or what may be termed the 
trailing lap of the plate. 

Let O D represent the position of the crank at the instant 
the main valve closes the port in the cylinder lace. (We shall 
fix this crank position O D presently, but for the moment we 
may assume it to be given or determined.) From the point 
z (in the semi-circle :^' z x) describe an arc N^ N, touching this 
crank radius O D. Now, if the distance c b were made equal 
to the radius of this arc we should have a reopening by the 
expansion plate at the precise instant that the main valve 
had closed the port. Taking, as before, a margin of safety 
of ^in., if we make the length or distance c 6 in the expansion 
plate Jin. in excess of the radius of arc N^ N, the desired 
margin of safety will be secured, and the crank radius O F 
drawn to touch arc L^ L (which is of Jin. greater radius than 
arc Ni N) will indicate the position of the crank when the slot 
harmlessly reopens. 

We have then, as the length A of each of the bars of the 
expansion plate, az + zc + cb, where a- -2- is the extreme 
plus lap, z c the width of slot in the valve, and c b the radius 
of an arc struck from centre z to touch a crank position OF 
later in the stroke than the position O D, at which the main 
valve closes. 

We have now to determine the width of the slot b d \n the 
plate, separating the two bars marked A (Fig. 2). This 
depends entirely upon the point in the stroke at which we 
wish the second or inner slot to begin its opening. As already 
mentioned in the earlier part of the present case, we assume 
that with the expansion plates screwed to their minimum 
distance apart, it is required that the inner slot shall begin 
to open when the crank is on the dead centre O B.'- 

To secure this result we have only to make the width of 
slot in the expansion plate equal to the radius of an arc struck 
from centre z, touching the crank position at which the slot is 
to begin its opening. In the present case we make the slot 
width b d equal to the radius of the arc S U, which touches 
crank position O B. The slot will continue to open until the 
crank reaches the position O J^, and will remain wide open 

* The first, or outer, slot opens very much earlier in the revolution, beginning 
when the crank occupied the position O w. (See Diagram 50a.) 



158 THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 

while the crank passes through the arc of revolution Ji z, 
after which both slots gradually close, cut-off occurring, of 
course, with the crank at O 5. This period Ji z (during which 
the inner slot remains wide open) is defined by the radius of 
an arc V^ V, struck from centre z, which corresponds with 
the distance 6 n in the expansion-plate slot, and is 3 units (or 
the width of the valve slot) less than the radius of arc S U. 

Supposing for a momeut that the width of slot in the plate 
were made 3 units, the same as the valve slots, then the inner 
slot would only begin to open when the crank reached the 
position O J, J being a radius of the crank circle drawn to 
touch the arc T^ T, struck from centre z with radius 3 units. 
In this case there would be no period like J^ z of full opening, 
but the inner slot would be opening during the period J z and 
closing during the period ^ 5 in the revolution of the crank. 

This opening and closing of the second or inner slot may 
perhaps be rendered clearer by a consideration of the small 
diagram 50a (which is exactly the same as Diagram 47a, ante), 
in which the horizontally shaded portion represents by its 
radial breadth the amount this inner slot is open at any crank 
position in the revolution, between O B, where opening begins, 
and O 5, where closure occurs, the vertical shading repre- 
senting the opening due to the jS.rst or outer slot. 




SCALE or UNITS 



Diagram 50a. 
Diagram 50a is derived from Diagram 50, and the lettering 
has the same significance, its only difference being that the 
arcs, yi V, Ti T, and S U are struck from centre O instead of 
from z, and that upon O x and O y, as diameters, the Zeuner 
circles are described. As the crank revolves we see the 



THE PROPORTIONS AND MOVEMENT OF SLIDE VALVES. 159 

breadth of horizontal shading, which is nothing when the 
crank is on the dead centre O B, increase gradually up to 3 
units at crank position O J^, remaining at 3 units breadth 
until the crank reaches the position O z, and then (the 
vertical shading from O z onward being common to both slots) 
gradually narrow to nothing again as the crank approaches 
the position of cut-off, O 5. 

After this digression, to demonstrate, by independent evi- 
dence, the correctness of our method, we may continue the 
construction of the valve and expansion plates. The total 
length of each plate is evidently made up of two bars (A A, 
Fig. 2) separated by a gap or slot whose width is equal to the 
radius of arc S U (Diagrams 50 or 50a). The length of bar 
A we have already determined. We have now to fix the mini- 
mum distance B separating the two inside edges c, c of the 
inner slots in the main valve. It is obvious that this is simply 
twice the length A of one of the bars in the expansion plate, 
the only condition being that the pair of plates, when screwed 
close together, shall have a minus lap of not less than 3 units, 
as prescribed. With the plates in this position, they are in 
effect identical with the fixed expansion plate Fig. 3, series C, 
illustrated on page 96. This distance B between the slots 
may of course be increased without disadvantage, provided 
there be room in the steam chest for the plates to travel. 

One more dimension of the main valve remains to be deter- 
mined, the distance C (Fig. 2 and Diagram 50). C is the 
distance separating the two slots, the outer and the inner, at 
either end of the valve, and is equal to the bar length A j^lus 
the distance b n or the radius of arc V^ V. This completes 
the dimensions of the valve and plates, and we may now 
proceed to construct a suitable main eccentric — that is, one 
which will give the proper steamway to correspond with the 
slot-opening for the latest specified point of cut-off. 

From the general centre O, lay off upon the base-line A B, 
to the right, O P equal to 4 units, the specified lap of the 
main valve, and with radius O P describe the lap (semi-) circle 
R G P. From the point P, lay off, still to the right, P Q equal 
to the given lead (1 unit), and at the point Q erect a per- 
pendicular to the base-line — the lead-line. The centre of the 
main eccentric will be somewhere on this line. If we find the 
amount of slot-opening when the crank is in the position 
corresponding to, say, fths of the stroke, and cause the main 
valve to have an opening equal to the combined opening of 
the two slots at this point, we shall have the full advantage 



160 THE PKOPORTIONS AND MOVEMENT OF SLIDE VALVES. 

of the slot-opening with no longer travel of the main valve 
than is just necessary to effect that purpose. 

To find the slot opening (the expansion plates being set for 
the maximum (fths) cut-off) when the crank is upon the 
radius O 3, corresponding to fths of the piston's stroke: — 
With centre O and radius O Z^ equal to 3 units (the prescribed 
extreme minus lap, giving the fths cutoff) describe a short 
arc Z^ t. Perpendicular to crank position O 3 produced below 
the axis O, draw the director Z^ to the point x. The slot is 
open a distance t Ty- when the crank occupies the position 3. 

Transfer douhl'e"^^ this distance t VJ to the exterior of the 
main lap circle R G P on the crank radius O 3 — i.e., lay off, 
upon O 3, G H equal to twice t Z-l Perpendicular to O 3 at H 
draw the director, and the point C, where it cuts the lead-line 
Q C, is the centre of the main eccentric to give the required 
port-opening at fths of the stroke. 

Join O C, which will be the throw, and A O C the angle 
with respect to the real position of the crank at A of the 
required main eccentric to fulfil the given conditions. With 
centre O and radius O C describe the main travel- circle 
A D C E. Touching the lap circle R G P, and at right angles 
to the radius O C, draw the chord D E of the travel-circle. 
Join O E (which will be the position of the crank when 
admission takes place) and O D (which will be its position 
when the main valve closes the port). 

Now make C X equal and parallel to O x, and join X. 
O X will be the throw, and A O X the angle with respect to 
the real position of the crank at A of the required expansion 
eccentric to fulfil the given conditions with the double-slotted 
valve and the expansion plates of Diagram 50. 



For a douhle-&lotte(\. valve. 



APR 20 1905 1 



